#6617560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524756);