#10087392

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
prime radical	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 9994588);