#6622244

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	#28485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(9,10,13,14)(11,16,12,15)(17,18,23,25)(19,24,27,31)(20,22,29,21)(26,28,32,30);(2,5)(6,8)(10,14)(11,12)(17,19,23,27)(18,24,25,31)(20,26,29,32)(21,30,22,28);(9,11,13,12)(10,15,14,16)(17,20,22,25,23,29,21,18)(19,26,28,31,27,32,30,24);(1,3,4,7)(2,6,5,8)(11,12)(15,16)(19,27)(24,31)(26,32)(28,30);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,20,25,29)(19,28,27,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=fh,e^2=hi,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, 6529440);