#7094678

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1281 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)(19,25,29,40)(20,26,34,41)(21,27,22,24)(30,39,43,48)(31,36,35,38)(32,37,33,42)(44,47,45,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,32)(21,22)(25,38)(27,28)(30,44)(33,34)(36,40)(37,42)(39,48)(43,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,35)(22,31)(23,29)(24,36)(26,39)(27,38)(28,40)(32,45)(33,44)(34,43)(37,46)(41,48)(42,47);(11,16)(13,15)(17,20,23,34)(18,26,28,41)(19,30,29,43)(21,33,22,32)(24,37,27,42)(25,39,40,48)(31,45,35,44)(36,46,38,47);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,31,29,35)(20,33,34,32)(25,38,40,36)(26,37,41,42)(30,45,43,44)(39,47,48,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,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=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=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, 7001874);