#8824456

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

4592 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(2,5)(6,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,38)(26,33,34,40)(27,35,32,37)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731652);