#8323177

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,f^2=gi,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, 8230373);