#6606521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(13,14)(15,24)(16,28)(17,22)(18,23)(19,34)(20,27)(21,32)(25,40)(26,37)(29,43)(30,39)(31,42)(33,36)(35,44)(38,41);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,27)(14,20,23,24)(16,29,31,25)(17,33,19,26)(21,38,39,35)(22,37,34,36)(28,40,42,43)(30,41,32,44);(6,9)(7,11)(13,16,17,30,18,31,19,32)(14,21,34,42,23,39,22,28)(15,25,33,44,27,29,26,41)(20,35,36,40,24,38,37,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,34)(15,26,27,33)(16,30,31,32)(20,36,24,37)(21,28,39,42)(25,41,29,44)(35,40,38,43);(6,9)(7,11)(15,27)(20,24)(25,29)(26,33)(35,38)(36,37)(40,43)(41,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513717);