#6595998

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

3174 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)(13,14,20,17)(15,16,25,28)(18,24,19,23)(21,22,27,26);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,24)(17,27)(18,22)(19,26)(20,25)(23,28);(6,9)(7,11)(13,16,20,28)(14,22,17,26)(15,23,25,24)(18,21,19,27);(5,7)(6,10)(8,11)(9,12)(13,17,20,14)(15,21,25,27)(16,22,28,26)(18,24,19,23);(6,9)(7,11)(14,17)(15,25)(22,26)(23,24)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6503194);