#7293171

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234321
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7200367);