#8720533

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

4337 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=ghi,e^2=h,f^2=h,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, 8627729);