#7471081

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

1109 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)(33,34,40,44,39,45,38,46)(35,47,55,59,50,41,49,58)(36,51,52,42,37,43,53,60)(48,61,63,62,54,57,64,56);(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)(34,41)(36,48)(37,54)(38,49)(39,50)(40,55)(42,56)(43,61)(44,58)(45,47)(46,59)(51,57)(52,63)(53,64)(60,62);(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31)(33,36,39,37)(34,42,45,60)(35,48,50,54)(38,52,40,53)(41,56,47,62)(43,44,51,46)(49,63,55,64)(57,59,61,58);(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)(33,37,39,36)(34,43,45,51)(35,48,50,54)(38,52,40,53)(41,57,47,61)(42,44,60,46)(49,64,55,63)(56,59,62,58);(2,8)(6,9)(7,16)(14,15)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)(33,38,39,40)(34,44,45,46)(35,49,50,55)(36,52,37,53)(41,58,47,59)(42,51,60,43)(48,63,54,64)(56,57,62,61)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7378277);