#10173532

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

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10080728);