#8422137

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

4480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329333);