#7092762

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,30,24)(20,25,33,40)(21,23,34,26)(29,42,44,38)(31,39,45,36)(32,41,35,37)(43,47,46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(21,34)(24,36)(25,40)(26,27)(28,39)(29,43)(31,45)(33,35)(38,42)(44,46);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,24,27,28)(20,29,33,44)(21,31,34,45)(23,36,26,39)(25,38,40,42)(32,43,35,46)(37,47,41,48);(11,16)(13,15)(17,20,22,33)(18,25,27,40)(19,29,30,44)(21,35,34,32)(23,37,26,41)(24,38,28,42)(31,46,45,43)(36,47,39,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(19,30)(24,28)(29,44)(31,45)(36,39)(38,42)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^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=eh,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, 6999958);