#8108772

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

4606 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)(3,7)(4,8)(5,6)(10,12)(14,16)(17,20)(21,23)(28,35)(32,37)(34,39)(38,40);(2,6)(3,8)(9,10,15,12)(11,16,19,14)(13,20,22,17)(18,21,24,23)(26,30)(27,33)(32,37)(34,39);(1,3)(2,4)(5,8)(6,7)(9,11,15,19)(10,16,12,14)(13,18,22,24)(17,23,20,21)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,10)(11,14,19,16)(13,20,22,17)(18,21,24,23);(1,4)(2,3)(5,7)(6,8)(9,13,15,22)(10,17,12,20)(11,18,19,24)(14,21,16,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,14)(10,11)(12,19)(13,21)(15,16)(17,18)(20,24)(22,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015968);