#10247619

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

4232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 10154815);