#10158597

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

4649 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^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, 10065793);