#6596529

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
prime radical	C₂×#3521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3016 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)(13,14,17,26,19,16,18,21)(15,22,27,20,24,28,25,23);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,23)(17,25)(18,27)(19,24)(21,28)(22,26);(6,9)(7,11)(13,16,19,14)(15,23,24,20)(17,26,18,21)(22,27,28,25);(5,7)(6,10)(8,11)(9,12)(15,24)(20,23)(22,28)(25,27);(6,9)(7,11)(13,17,19,18)(14,21,16,26)(15,25,24,27)(20,28,23,22)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6503725);