#8716784

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

4598 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8623980);