#10089008

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

4387 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=g,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, 9996204);