#7471090

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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

1111 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31)(33,34,39,43,38,44,37,45)(35,46,48,57,49,40,50,62)(36,42,52,41,53,61,54,59)(47,60,64,58,51,56,63,55);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,28)(18,23,25,26)(20,27,21,29)(24,31,30,32)(33,35,38,49)(34,40,44,46)(36,51,53,47)(37,50,39,48)(41,58,59,55)(42,60,61,56)(43,62,45,57)(52,63,54,64);(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31)(33,36,38,53)(34,41,44,59)(35,47,49,51)(37,52,39,54)(40,55,46,58)(42,45,61,43)(48,63,50,64)(56,57,60,62);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)(33,36)(34,42)(35,47)(37,54)(38,53)(39,52)(40,56)(41,43)(44,61)(45,59)(46,60)(48,64)(49,51)(50,63)(55,62)(57,58);(2,8)(6,9)(7,16)(14,15)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)(33,37,38,39)(34,43,44,45)(35,48,49,50)(36,52,53,54)(40,57,46,62)(41,42,59,61)(47,63,51,64)(55,60,58,56)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7378286);