#7095128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,32,43)(20,25,34,41)(21,27,22,26)(24,36,39,31)(30,45,33,48)(35,44,37,42)(38,46,40,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,36,23,32,22,31)(18,24,26,43,28,39,27,29)(20,33,37,47,34,30,35,46)(25,40,44,45,41,38,42,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,28,41)(19,30,32,33)(21,37,22,35)(24,38,39,40)(26,44,27,42)(29,45,43,48)(31,47,36,46);(11,16)(13,15)(19,31)(20,34)(21,22)(24,29)(25,41)(26,27)(30,46)(32,36)(33,47)(38,48)(39,43)(40,45);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,35,34,37)(24,29,39,43)(25,42,41,44)(30,46,33,47)(38,48,40,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=gh,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7002324);