#6445714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2922 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352910);