#512781

order	512
nilpotency class	5
isoclinism	isoclinic class of #512779
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2177
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2263	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

429 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,18,45)(4,21)(5,10)(6,26)(7,23)(8,13)(9,17,35,52)(11,30)(12,25)(15,47,34,49)(16,43,20,48)(19,55,54,62)(22,57)(24,59)(27,40,53,44)(28,61)(29,41)(31,42)(32,46,50,38)(33,37)(36,60,64,63)(39,51)(56,58);(1,3)(2,9)(4,15)(5,16)(6,27)(7,32)(8,18)(10,34)(11,38)(12,40)(13,35)(14,42)(17,51)(19,25)(20,21)(22,55)(23,36)(24,60)(26,45)(28,62)(29,54)(30,52)(31,53)(33,50)(37,64)(39,46)(41,44)(43,58)(47,61)(48,59)(49,57)(56,63);(1,4,21,2,8,10,5,13)(3,15,20,9,18,34,16,35)(6,28,56,39,31,22,24,11)(7,23,29,12,33,37,25,41)(14,17,43,47,45,52,48,49)(19,44,32,36,54,40,50,64)(26,30,59,57,42,51,58,61)(27,62,63,46,53,55,60,38);(1,5,8,21)(2,10,13,4)(3,16,18,20)(6,24,31,56)(7,25,33,29)(9,34,35,15)(11,28,39,22)(12,37,41,23)(14,43,45,48)(17,49,52,47)(19,50,54,32)(26,59,42,58)(27,60,53,63)(30,61,51,57)(36,40,64,44)(38,62,46,55)

Polycyclic presentation:

2^9|a^2=f,c^2=fhi,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224987, #72224997, #72224998, …
maximal normal subgroup of	#72224987, #72224997, #72224998, …
maximal quotient of	#72232009, #72232010, #72232019, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419977);