#8936959

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

4612 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8844155);