#6573422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

3164 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)(13,14,18,22,19,23,17,21)(15,24,25,16,26,27,28,20);(5,6,10,9)(7,12,8,11)(13,15)(14,20)(16,23)(17,25)(18,28)(19,26)(21,24)(22,27);(2,4)(5,7,10,8)(6,11,9,12)(13,16,19,20)(14,15,23,26)(17,24,18,27)(21,28,22,25);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,21,23,22)(15,25,26,28)(16,27,20,24);(5,9,10,6)(7,12,8,11)(13,17,19,18)(14,22,23,21)(15,25,26,28)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480618);