#10325500

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

4515 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232696);