#8912578

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

4491 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)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,31,36)(27,37,40,32)(28,39,38,33)(29,34,35,30);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27,31,40)(26,32,36,37)(28,35,38,29)(30,39,34,33);(1,3)(2,4)(5,6)(7,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,28,31,38)(26,33,36,39)(27,35,40,29)(30,37,34,32);(3,6)(4,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,40)(30,32)(31,38)(34,37)(36,39);(1,4,5,8)(2,6,7,3)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,35)(26,34,36,30)(27,38,40,28)(32,39,37,33);(1,2)(3,8)(4,6)(5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30,31,34)(26,35,36,29)(27,39,40,33)(28,32,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819774);