#8094954

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

4467 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)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2,5,3)(4,6,8,7)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(2,3)(6,7)(10,15)(14,21)(16,19)(22,23)(25,27,31,28)(26,33,35,32)(29,38,34,36)(30,37,40,39);(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28,31,27)(26,33,35,32)(29,38,34,36)(30,39,40,37);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,29,31,34)(26,30,35,40)(27,36,28,38)(32,37,33,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,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, 8002150);