#6607841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#24790	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

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, 6515037);