#7089874

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	#31021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,35,46)(20,26,21,27)(22,28,23,39)(25,33,42,34)(31,41,37,48)(32,40,47,45)(36,43,38,44);(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,37)(22,34)(23,33)(24,35)(26,40)(27,45)(28,46)(29,42)(30,39)(32,38)(36,47)(41,43)(44,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,38)(23,36)(25,40)(27,29)(28,44)(30,41)(32,34)(33,47)(35,37)(39,43)(42,45)(46,48);(11,16)(13,15)(17,21,24,20)(18,27,29,26)(19,32,35,47)(22,38,23,36)(25,41,42,48)(28,44,39,43)(30,40,46,45)(31,33,37,34);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,33,35,34)(20,36,21,38)(25,30,42,46)(26,43,27,44)(31,32,37,47)(40,48,45,41)

Polycyclic presentation:

2^9|a^2=i,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=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, 6997070);