#6596546

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×#3522	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3099 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)(13,14,16,22,18,21,17,19)(15,23,25,20,26,27,24,28);(2,4)(5,6,8,9)(7,10,11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(6,9)(7,11)(13,15,18,26)(14,20,21,28)(16,24,17,25)(19,27,22,23);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,19,21,22)(15,24,26,25)(20,27,28,23);(6,9)(7,11)(14,21)(19,22)(20,28)(23,27)

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=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, 6503742);