#417629

order	512
nilpotency class	3
isoclinism	isoclinic class of #417628
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2307 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)(3,9)(4,10)(5,22)(6,12)(7,28)(8,13)(11,24)(14,41)(15,45)(16,31)(17,32)(18,49)(19,51)(20,36)(21,37)(23,39)(25,40)(26,33)(27,38)(29,43)(30,47)(34,44)(35,54)(42,59)(46,58)(48,57)(50,63)(52,56)(53,62)(55,61)(60,64);(1,3)(2,9)(4,18)(5,15)(6,26)(7,17)(8,16)(10,34)(11,30)(12,33)(13,31)(14,21)(19,52)(20,44)(22,45)(23,48)(24,47)(25,46)(27,43)(28,32)(29,37)(35,60)(36,49)(38,41)(39,57)(40,58)(42,54)(50,53)(51,63)(55,64)(56,62)(59,61);(1,4)(2,10)(3,14)(5,19)(6,27)(7,21)(8,20)(9,29)(11,35)(12,38)(13,36)(15,42)(16,43)(17,44)(18,26)(22,51)(23,55)(24,54)(25,53)(28,37)(30,56)(31,41)(32,49)(33,34)(39,61)(40,62)(45,60)(46,64)(47,50)(48,52)(57,63)(58,59);(1,5,8,25)(2,11,13,39)(3,15,16,46)(4,19,20,53)(6,24,7,23)(9,30,31,57)(10,35,36,61)(12,40,28,22)(14,42,43,64)(17,48,26,47)(18,50,44,52)(21,55,27,54)(29,56,41,63)(32,45,33,58)(34,59,49,60)(37,51,38,62)

Polycyclic presentation:

2^9|d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287064, #72287286, #72287292
maximal normal subgroup of	#72287064, #72287286, #72287292

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324825);