#484232

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1595
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1083 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1083 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,15)(4,20)(5,11)(6,26)(7,13)(8,14)(9,18)(10,23)(12,29)(16,45)(17,43)(19,44)(21,37)(22,56)(24,40)(25,39)(27,33)(28,36)(30,41)(31,38)(32,59)(34,51)(35,52)(42,48)(46,64)(47,53)(49,62)(50,60)(54,58)(55,63)(57,61);(1,3)(2,9)(4,16)(5,17)(6,27)(7,31)(8,19)(10,32)(11,33)(12,38)(13,34)(14,35)(15,26)(18,29)(20,53)(21,46)(22,57)(23,48)(24,49)(25,52)(28,59)(30,51)(36,62)(37,61)(39,43)(40,60)(41,44)(42,54)(45,56)(47,58)(50,55)(63,64);(1,4)(2,10)(3,16)(5,21)(6,28)(7,23)(8,24)(9,32)(11,20)(12,36)(13,40)(14,37)(15,42)(17,46)(18,50)(19,49)(22,30)(25,58)(26,54)(27,59)(29,55)(31,48)(33,53)(34,60)(35,61)(38,62)(39,56)(41,63)(43,45)(44,64)(47,52)(51,57);(1,5,8,7)(2,11,14,13)(3,17,19,31)(4,21,24,23)(6,25,30,29)(9,33,35,34)(10,20,37,40)(12,26,39,41)(15,43,44,38)(16,46,49,48)(18,27,52,51)(22,55,28,58)(32,53,61,60)(36,54,56,63)(42,45,64,62)(47,57,50,59)

Polycyclic presentation:

2^9|d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213991, #72217918, #72307392, …
maximal normal subgroup of	#72217918, #72307392, #72307417, …
maximal quotient of	#72213777, #72213794, #72213801

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391428);