#6494364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2514 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)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,32)(26,34)(28,35)(29,36)(31,38)(33,37)(39,42)(40,43)(41,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,27,19,28,18,29)(14,20,23,25,24,36,22,35)(16,26,31,40,33,41,32,42)(21,34,37,39,38,44,30,43);(6,9)(10,12)(13,16,17,31,19,33,18,32)(14,21,23,37,24,38,22,30)(15,26,27,40,28,41,29,42)(20,34,25,39,36,44,35,43);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,40,41,42)(34,43,44,39);(5,7,8,11)(6,10,9,12)(15,28)(20,36)(25,35)(26,41)(27,29)(34,44)(39,43)(40,42)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6401560);