#6627362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

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, 6534558);