#7101273

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,26)(21,28)(22,27)(23,29)(24,32)(25,33)(31,43)(34,40)(35,42)(36,46)(37,45)(38,41)(39,44)(47,48);(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,29,41)(20,34,37,47)(21,32,22,38)(25,27,30,28)(26,43,45,42)(31,36,35,39)(40,44,48,46);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,28)(26,43)(27,41)(29,30)(31,37)(34,36)(39,47)(40,44)(42,45)(46,48);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,31)(21,36)(22,39)(23,37)(24,40)(25,42)(27,44)(28,46)(29,45)(30,43)(32,34)(33,35)(38,47)(41,48);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,32,33,38)(20,36,37,39)(24,30,41,25)(26,44,45,46)(31,34,35,47)(40,43,48,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gi,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=ehi,f^b=fi,d^c=dh,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, 7008469);