#8517333

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8424529);