#10089005

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

4688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996201);