#7623669

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623668
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285648
maximal normal subgroup of	#72285648
maximal quotient of	#72342601, #72342603

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530865);