#10180998

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

4523 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(26,30)(28,36)(31,33)(34,40);(1,2,5,4)(3,6,8,7)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,22,23,18)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,7)(5,8)(10,16)(11,19)(12,21)(18,22)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,3,5,8)(2,6,4,7)(9,12)(10,17)(11,20)(13,19)(14,22)(15,21)(16,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,20)(10,18,16,22)(11,21,19,12)(14,23,24,17)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,16)(12,22)(13,17)(15,24)(18,21)(20,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=i,f^2=g,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, 10088194);