#8093819

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

4307 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=g,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, 8001015);