#6993016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30463	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,32)(21,26)(22,24)(23,30)(25,34)(27,36)(29,38)(31,39)(33,41)(35,42)(37,44)(40,46)(43,47)(45,48);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,33)(21,30)(22,31)(24,37)(25,41)(26,28)(27,39)(29,34)(32,38)(35,45)(36,44)(40,48)(42,47)(43,46);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(18,24)(20,34)(22,36)(23,37)(25,42)(26,27)(28,39)(29,33)(31,44)(32,40)(38,48)(41,47);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(17,20,21,34)(18,25,26,32)(19,29,30,33)(22,35,36,46)(23,38,28,41)(24,40,27,42)(31,43,44,45)(37,47,39,48);(1,6,7,5)(2,3,8,4)(18,26)(19,30)(20,34)(24,27)(31,44)(35,46)(38,41)(47,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900212);