#7063508

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

Polycyclic presentation:

2^9|a^2=fg,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, 6970704);