#8588926

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

4431 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=g,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, 8496122);