#7507113

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(17,18,21,23)(19,27,30,24)(20,25,31,35)(22,26,33,36)(28,37,34,39)(29,41,44,38)(32,40,45,46)(42,47,43,48);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,31)(22,33)(24,37)(27,39)(29,43)(30,34)(38,48)(40,46)(41,47)(42,44);(10,14)(13,16)(17,19,21,30)(18,24,23,27)(20,29,31,44)(22,34,33,28)(25,38,35,41)(26,37,36,39)(32,43,45,42)(40,47,46,48);(2,7)(3,8)(10,14)(13,16)(17,20)(18,25)(19,29)(21,31)(22,32)(23,35)(24,38)(26,40)(27,41)(28,42)(30,44)(33,45)(34,43)(36,46)(37,47)(39,48);(1,4,5,6)(2,3,7,8)(18,23)(19,30)(20,31)(26,36)(28,34)(32,45)(38,41)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414309);