#152286

order	512
nilpotency class	5
isoclinism	isoclinic class of #152284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12813
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224991, #72225005, #72225006, …
maximal normal subgroup of	#72224991, #72225005, #72225006, …
maximal quotient of	C₂×#152286

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59482);