#7590118

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497314);