#8233424

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

4510 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=g,f^2=i,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, 8140620);