#8112518

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

4618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019714);