#8509518

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

4358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416714);