#8214379

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

4416 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)(11,20)(12,18)(13,14)(15,17)(16,19)(21,22)(23,24)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,16,15)(10,17,19,20)(12,21,14,24)(13,22,18,23)(26,28)(27,33)(32,36)(35,39);(2,5)(6,7)(9,12,16,14)(10,18,19,13)(11,21,15,24)(17,23,20,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(9,13)(10,12)(11,22)(14,19)(15,23)(16,18)(17,21)(20,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3,4,8)(2,6,5,7)(9,14,16,12)(10,13,19,18)(11,21,15,24)(17,23,20,22)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,11)(10,17,19,20)(12,21,14,24)(13,23,18,22)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=gi,f^2=g,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, 8121575);