#7508547

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,32,40)(20,33,35,25)(21,26,22,27)(29,37,31,39)(30,44,46,38)(34,42,36,41)(43,48,45,47);(1,3)(2,6)(4,7)(5,8)(17,19,23,32)(18,24,28,40)(20,30,35,46)(21,31,22,29)(25,38,33,44)(26,39,27,37)(34,43,36,45)(41,47,42,48);(9,11,12,16)(10,13,14,15)(19,29)(21,22)(24,37)(26,27)(30,43)(31,32)(34,36)(38,47)(39,40)(41,42)(44,48)(45,46);(2,7)(3,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,38)(26,42)(27,41)(28,33)(29,43)(31,45)(32,46)(37,47)(39,48)(40,44);(1,4,5,6)(2,3,7,8)(17,21,23,22)(18,26,28,27)(19,31,32,29)(20,34,35,36)(24,39,40,37)(25,41,33,42)(30,43,46,45)(38,47,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,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, 7415743);