#8126033

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8033229);