#8726722

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

4335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633918);