#7507233

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,39)(28,38)(29,34)(30,35)(31,40)(32,36)(33,37)(41,48)(42,47)(43,45)(44,46);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,30)(22,32)(24,37)(28,42)(29,33)(34,39)(38,48)(40,46)(41,43)(45,47);(10,14)(13,16)(17,19,21,29)(18,24,23,34)(20,28,30,43)(22,33,32,27)(25,38,35,45)(26,37,36,39)(31,42,44,41)(40,47,46,48);(2,7)(3,8)(10,14)(13,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,35)(24,38)(26,40)(27,41)(29,43)(32,44)(33,42)(34,45)(36,46)(37,47)(39,48);(1,4,5,6)(2,3,7,8)(19,29)(20,30)(24,34)(25,35)(27,33)(31,44)(37,39)(40,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414429);