#8814033

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

4587 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8721229);