#7510722

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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)(9,10,12,13)(11,15,16,14)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,33)(30,38)(31,41)(32,44)(34,39)(35,42)(36,40)(37,43)(45,47)(46,48);(1,3)(2,6)(4,7)(5,8)(17,19,22,33)(18,24,25,29)(20,34,36,31)(21,32,23,30)(26,41,40,39)(27,38,28,44)(35,46,37,45)(42,47,43,48);(10,13)(14,15)(18,25)(19,30)(21,23)(24,38)(26,40)(29,44)(31,45)(32,33)(34,46)(35,37)(39,47)(41,48);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,20,22,36)(18,26,25,40)(19,31,33,34)(21,37,23,35)(24,39,29,41)(27,43,28,42)(30,45,32,46)(38,47,44,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,28)(19,32,33,30)(20,35,36,37)(24,38,29,44)(26,42,40,43)(31,45,34,46)(39,48,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7417918);