#6596306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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

2995 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)(13,14,18,23,19,22,24,28)(15,21,16,26,25,17,27,20);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,25)(14,20,22,26)(16,18,27,24)(17,28,21,23);(6,9)(7,11)(13,16,19,27)(14,21,22,17)(15,24,25,18)(20,28,26,23);(5,7)(6,10)(8,11)(9,12)(13,17,19,21)(14,16,22,27)(15,23,25,28)(18,20,24,26);(6,9)(7,11)(14,22)(17,21)(20,26)(23,28)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503502);