#6454800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2990 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 6361996);