#8682231

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

4543 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)(9,10,15,18)(11,19,12,16)(13,14,17,24)(20,23,22,21)(26,30)(27,33)(32,35)(34,36);(3,7)(6,8)(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(26,30)(31,38)(32,35)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,16,18,19)(13,22,17,20)(14,21,24,23)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,14)(11,20)(12,22)(15,17)(16,21)(18,24)(19,23)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8589427);