#8747937

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8655133);