#8214972

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

4594 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:

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

Polycyclic presentation:

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

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, 8122168);