#8722942

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

4454 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 8630138);