#136017

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
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	#8854
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,15,19,43)(4,20)(5,24)(6,13)(7,12)(8,14)(9,17,34,51)(10,22)(11,27)(16,44,47,62)(18,52,31,38)(21,55)(23,40)(25,35,50,33)(26,36)(28,41)(29,37)(30,39)(32,59,61,56)(42,48,64,46)(45,60,49,53)(54,58)(57,63);(1,3,7,31,8,19,6,18)(2,9,13,33,14,34,12,35)(4,16,22,46,23,47,29,48)(5,25,27,17,28,50,30,51)(10,32,20,53,37,61,40,60)(11,38,24,15,39,52,41,43)(21,56,58,45,26,59,57,49)(36,62,54,42,55,44,63,64);(1,4)(2,10)(3,16)(5,26)(6,29)(7,22)(8,23)(9,32)(11,36)(12,40)(13,20)(14,37)(15,42)(17,49)(18,48)(19,47)(21,28)(24,54)(25,59)(27,57)(30,58)(31,46)(33,53)(34,61)(35,60)(38,62)(39,55)(41,63)(43,64)(44,52)(45,51)(50,56)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217918
maximal normal subgroup of	#72217918
maximal quotient of	#72217029, #72217037, #72217040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43213);