#8094949

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002145);