#7228645

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7135841);