#10041016

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

4565 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)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,31,29)(27,33,35,28)(30,37,38,32)(34,36,40,39);(1,2)(3,8)(4,7)(5,6)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,31,35)(26,28,29,33)(30,34,38,40)(32,36,37,39);(2,6)(3,7)(10,15)(14,21)(16,19)(22,23)(26,29)(28,33)(32,37)(36,39);(1,2,5,6)(3,4,7,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,31,26)(27,28,35,33)(30,32,38,37)(34,39,40,36);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,30)(26,32)(27,34)(28,36)(29,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=g,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, 9948212);