#6513128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3059 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)(13,14,18,23,19,22,17,30)(15,24,26,34,27,35,32,20)(16,21,28,38,29,36,31,37)(25,39,41,43,42,44,40,33);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,22,34)(16,25,29,42)(17,26,18,32)(21,33,36,43)(23,35,30,24)(28,40,31,41)(37,39,38,44);(2,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,28)(19,29)(20,33)(22,36)(23,38)(24,39)(26,41)(27,42)(30,37)(32,40)(34,43)(35,44);(5,7,8,11)(6,10,9,12)(14,22)(20,34)(21,36)(23,30)(24,35)(33,43)(37,38)(39,44);(6,9)(10,12)(13,17,19,18)(14,23,22,30)(15,26,27,32)(16,28,29,31)(20,35,34,24)(21,37,36,38)(25,40,42,41)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420324);