#6595939

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

3225 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)(15,16)(17,20)(18,23)(19,24)(21,25)(22,27)(26,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15,20,26)(14,21,17,27)(16,23,28,24)(18,25,19,22);(6,9)(7,11)(13,16)(14,22)(15,24)(17,25)(18,27)(19,21)(20,28)(23,26);(5,7)(6,10)(8,11)(9,12)(13,17,20,14)(15,21,26,27)(16,22,28,25)(18,23,19,24);(6,9)(7,11)(13,18,20,19)(14,23,17,24)(15,22,26,25)(16,27,28,21)

Polycyclic presentation:

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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503135);