#6573441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

3171 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)(13,14,19,22,18,15,16,20)(17,21,25,28,26,23,24,27);(5,6,10,9)(7,12,8,11)(13,15,18,14)(16,20,19,22)(17,23,26,21)(24,27,25,28);(2,4)(5,7,10,8)(6,11,9,12)(13,16,18,19)(14,20,15,22)(17,25,26,24)(21,28,23,27);(5,8,10,7)(6,11,9,12)(13,17,18,26)(14,21,15,23)(16,24,19,25)(20,27,22,28);(5,9,10,6)(7,12,8,11)(17,26)(21,23)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,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, 6480637);