#8096956

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

4887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004152);