#9105316

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=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, 9012512);