#8891593

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

4297 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:

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

Polycyclic presentation:

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

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, 8798789);