#6999915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(11,13)(12,14)(15,16)(17,18,22,28)(19,29,33,24)(20,26,35,43)(21,27,23,38)(25,37,44,34)(30,40,31,45)(32,42,36,41)(39,48,46,47);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,36)(22,33)(23,32)(25,39)(26,45)(27,42)(28,29)(31,35)(34,48)(37,47)(38,41)(40,43)(44,46);(9,11)(10,13)(12,16)(14,15)(17,20,22,35)(18,25,28,44)(19,30,33,31)(21,34,23,37)(24,39,29,46)(26,27,43,38)(32,47,36,48)(40,41,45,42);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,34)(24,40)(25,38)(27,44)(28,43)(29,45)(30,33)(32,47)(36,48)(39,42)(41,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,36)(20,34,35,37)(24,41,29,42)(25,43,44,26)(30,47,31,48)(39,45,46,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6907111);