#7337170

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(4,7)(5,8)(17,18,22,24)(19,29,30,40)(20,26,35,38)(21,28,23,27)(25,32,37,33)(31,43,44,48)(34,42,36,41)(39,46,47,45);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(26,38)(27,28)(31,44)(32,33)(41,42)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,21,32,22,30,23,33)(18,25,27,40,24,37,28,29)(20,31,36,46,35,44,34,45)(26,39,42,48,38,47,41,43);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20)(18,26)(19,31)(21,36)(22,35)(23,34)(24,38)(25,39)(27,42)(28,41)(29,43)(30,44)(32,46)(33,45)(37,47)(40,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,34,35,36)(25,40,37,29)(26,41,38,42)(31,45,44,46)(39,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7244366);