#8425451

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

4639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332647);