#136267

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8873
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,15,18,46)(4,20,22,57)(5,24)(6,28)(7,13)(8,14)(9,17,36,54)(10,21,39,59)(11,27)(12,31)(16,48)(19,47,33,42)(23,43,34,58)(25,37,29,38)(26,40,30,41)(32,44)(35,52)(45,49)(50,56)(51,61)(53,63)(55,62)(60,64);(1,3,7,33,8,18,31,19)(2,9,12,38,14,36,13,37)(4,16,34,51,22,52,23,55)(5,25,27,17,6,29,32,54)(10,35,41,61,39,48,40,62)(11,42,24,15,44,47,28,46)(20,56,43,63,57,45,58,60)(21,53,30,50,59,64,26,49);(1,4,8,22)(2,10)(3,16,18,52)(5,26,6,30)(7,34,31,23)(9,35)(11,43)(12,41)(13,40)(14,39)(15,45)(17,53,54,64)(19,55,33,51)(20,28)(21,32,59,27)(24,57)(25,49,29,50)(36,48)(37,62)(38,61)(42,63)(44,58)(46,56)(47,60)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gh,d^2=g,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214726, #72217918
maximal normal subgroup of	#72217918
maximal quotient of	#72214470, #72214486, #72214501, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43463);