#6596411

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3237 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)(13,14,19,23,18,24,17,22)(15,25,26,20,21,28,27,16);(5,6,8,9)(7,10,11,12)(13,15,18,21)(14,20,24,16)(17,26,19,27)(22,28,23,25);(2,4)(6,9)(7,11)(13,16,18,20)(14,21,24,15)(17,28,19,25)(22,26,23,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,24,23)(15,26,21,27)(16,25,20,28);(6,9)(7,11)(13,17,18,19)(14,23,24,22)(15,26,21,27)(16,28,20,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=i,e^2=i,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, 6503607);