#10023305

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9930501);