#8943992

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

4683 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851188);