#6596586

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

3035 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)(13,14,16,21,18,22,17,20)(15,23,24,28,25,27,26,19);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,25)(14,19,22,28)(16,26,17,24)(20,23,21,27);(6,9)(7,11)(13,14,18,22)(15,19,25,28)(16,20,17,21)(23,24,27,26);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,28,23);(6,9)(7,11)(15,25)(19,28)(23,27)(24,26)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,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, 6503782);