#7590126

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	#23866	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,24,20,27)(21,29,31,25)(26,30,28,32);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,22)(21,30)(23,27)(25,26)(28,29)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,19)(18,24,23,27)(21,28,31,26)(25,32,29,30);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,18,22,23)(19,24,20,27)(21,25,31,29)(26,32,28,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,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, 7497322);