#7232003

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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

1124 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)(3,10,13,7)(4,8,5,14)(11,16,12,15)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34)(35,43)(36,41)(37,51)(38,39)(40,46)(42,44)(45,58)(47,57)(48,53)(49,56)(50,55)(52,59)(54,62)(60,64)(61,63);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,31)(24,32)(26,29)(27,30)(34,39)(36,48)(40,52)(42,51)(43,57)(45,60)(49,50)(61,62);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,23)(19,25)(24,27)(26,28)(33,35,44,59,38,47,37,46)(34,40,51,57,39,52,42,43)(36,45,49,61,48,60,50,62)(41,54,55,64,53,63,56,58);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,24,23,27)(19,26,25,28)(22,30,31,32)(33,36)(34,41)(35,45)(37,50)(38,48)(39,53)(40,54)(42,56)(43,58)(44,49)(46,62)(47,60)(51,55)(52,63)(57,64)(59,61);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,31)(26,28)(30,32)(33,37,38,44)(34,42,39,51)(35,46,47,59)(36,49,48,50)(40,43,52,57)(41,55,53,56)(45,61,60,62)(54,64,63,58)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7139199);