#6617635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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

3045 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)(13,14,18,23,19,24,17,22)(15,25,26,16,21,27,28,20);(6,10)(8,11)(13,15,19,21)(14,20,24,16)(17,26,18,28)(22,25,23,27);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,19,20)(14,21,24,15)(17,25,18,27)(22,26,23,28);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,24,23)(15,26,21,28)(16,27,20,25);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,23,24,22)(15,26,21,28)(16,25,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524831);