#8649023

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

4422 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556219);