#7091282

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1258 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:

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

Polycyclic presentation:

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

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, 6998478);