#8633712

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

4580 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,f^2=hi,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, 8540908);