#8114341

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

4536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021537);