#7063524

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 6970720);