#7471155

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1108 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)(33,34,39,44,40,45,38,54)(35,41,48,57,49,58,47,62)(36,50,51,59,37,43,53,42)(46,56,61,64,52,60,63,55);(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)(34,41)(36,46)(37,52)(38,47)(39,48)(40,49)(42,55)(43,60)(44,57)(45,58)(50,56)(51,61)(53,63)(54,62)(59,64);(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31)(33,36,40,37)(34,42,45,59)(35,46,49,52)(38,51,39,53)(41,55,58,64)(43,54,50,44)(47,61,48,63)(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)(33,37,40,36)(34,43,45,50)(35,46,49,52)(38,51,39,53)(41,56,58,60)(42,54,59,44)(47,63,48,61)(55,57,64,62);(2,8)(6,9)(7,16)(14,15)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)(33,38,40,39)(34,44,45,54)(35,47,49,48)(36,51,37,53)(41,57,58,62)(42,43,59,50)(46,61,52,63)(55,60,64,56)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7378351);