#8323162

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

4461 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)(3,5)(4,7)(6,8)(25,26,31,34)(27,33,29,32)(28,37,38,30)(35,36,39,40);(2,3)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27,31,29)(26,32,34,33)(28,35,38,39)(30,36,37,40);(1,2,5,3)(4,6,8,7)(10,15)(13,23)(16,21)(17,24)(25,27,31,29)(26,33,34,32)(28,35,38,39)(30,40,37,36);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,40)(34,37);(1,4)(2,6)(3,7)(5,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,29,31,27)(26,32,34,33)(28,35,38,39)(30,40,37,36);(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230358);