#7138873

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=hi,e^2=f,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, 7046069);