#6494323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401519);