#6494329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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

2456 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)(7,11)(10,12)(13,14,18,21)(15,23,27,40)(16,20,28,37)(17,30,31,22)(19,32,35,26)(24,38,25,39)(29,44,41,36)(33,43,34,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,26,18,27,31,32)(14,19,22,23,21,35,30,40)(16,25,29,43,28,24,41,42)(20,34,36,39,37,33,44,38);(1,2,3,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,37)(22,36)(23,38)(25,27)(26,42)(30,44)(31,41)(32,43)(34,35)(39,40);(6,9)(10,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,34)(21,37)(22,36)(23,39)(24,27)(26,43)(30,44)(31,41)(32,42)(33,35)(38,40);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,35)(22,30)(23,40)(24,25)(29,41)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401525);