#512702

order	512
nilpotency class	5
isoclinism	isoclinic class of #512698
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#357
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#3415	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224987, #72225009, #72225010, …
maximal normal subgroup of	#72224987, #72225009, #72225010, …
maximal quotient of	#72232008, #72232018, #72234331, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419898);