#7033532

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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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)(9,10,13,16)(11,12,15,14)(17,18)(19,29)(20,26)(21,27)(22,28)(23,25)(24,32)(30,42)(31,38)(33,40)(34,39)(35,44)(36,41)(37,43)(45,48)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,32)(18,24,28,29)(20,34,35,30)(21,31,23,33)(25,38,27,40)(26,42,44,39)(36,46,37,45)(41,48,43,47);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(18,25)(20,35)(21,23)(24,38)(26,43)(27,28)(29,40)(30,34)(31,33)(39,48)(41,44)(42,47);(2,5)(6,8)(11,15)(12,14)(17,20)(18,26)(19,30)(21,37)(22,35)(23,36)(24,39)(25,41)(27,43)(28,44)(29,42)(31,46)(32,34)(33,45)(38,47)(40,48);(10,16)(11,15)(17,21,22,23)(18,27,28,25)(19,31,32,33)(20,36,35,37)(24,40,29,38)(26,41,44,43)(30,45,34,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 6940728);