#8799413

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

4360 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:

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

Polycyclic presentation:

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

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, 8706609);