#494051

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20733
prime radical	#3584	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

907 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224958, …
maximal normal subgroup of	#72217287, #72217290, #72224958, …
maximal quotient of	C₂×#494051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401247);