#7092057

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

1358 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)(18,24)(19,28)(21,22)(25,40)(26,32)(27,37)(29,39)(30,33)(34,35)(41,45)(42,48)(43,44);(9,10)(11,15)(12,14)(13,16)(17,18,21,37,23,27,22,24)(19,29,38,39,28,25,31,40)(20,32,34,44,36,43,35,26)(30,45,47,41,33,48,46,42);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,33)(21,38)(22,31)(23,28)(24,39)(26,42)(27,29)(30,36)(32,45)(34,46)(35,47)(37,40)(41,44)(43,48);(11,16)(13,15)(17,20,23,36)(18,26,27,44)(19,30,28,33)(21,35,22,34)(24,32,37,43)(25,41,29,42)(31,47,38,46)(39,48,40,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,24,27,37)(19,31,28,38)(20,34,36,35)(25,40,29,39)(26,43,44,32)(30,46,33,47)(41,48,42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gh,d^2=i,e^2=g,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, 6999253);