#6607924

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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)(17,18,22,20)(19,26,29,32)(21,25,23,31)(24,28,27,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,24)(20,22)(21,31)(23,25)(26,30)(27,29)(28,32);(11,12)(14,16)(17,19,21,28,22,29,23,30)(18,24,31,32,20,27,25,26);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,20,22,18)(19,27,29,24)(21,25,23,31)(26,30,32,28);(11,12)(14,16)(17,21,22,23)(18,25,20,31)(19,28,29,30)(24,26,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=f,e^2=i,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^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6515120);