#8599630

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

4563 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)(11,16)(12,18)(13,17)(14,15)(19,23)(20,24)(21,22)(25,26,32,36)(27,30,31,33)(28,35,29,34)(37,38,39,40);(2,5)(6,8)(10,15)(16,18)(17,22)(23,24)(25,27,32,31)(26,33,36,30)(28,37,29,39)(34,38,35,40);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,16,15,18)(13,19,21,20)(17,23,22,24)(25,28,32,29)(26,34,36,35)(27,37,31,39)(30,40,33,38);(3,7)(6,8)(9,12,14,11)(10,16,15,18)(13,19,21,20)(17,24,22,23)(25,29,32,28)(26,35,36,34)(27,37,31,39)(30,40,33,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,30,32,33)(26,27,36,31)(28,38,29,40)(34,39,35,37);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,17,15,22)(11,19,12,20)(16,23,18,24)(25,31,32,27)(26,33,36,30)(28,39,29,37)(34,38,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8506826);