#10173524

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	#47826	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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)(3,4)(5,7)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,33)(27,32,36,40)(28,29,34,39)(30,38,37,35);(1,3)(2,6)(4,7)(5,8)(12,19)(15,17)(20,24)(22,23)(25,27,31,36)(26,32,33,40)(28,37,34,30)(29,35,39,38);(1,2,5,7)(3,6,8,4)(10,11)(15,17)(16,18)(22,23)(26,33)(27,36)(28,34)(35,38);(2,7)(4,6)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,29)(27,37)(30,36)(31,34)(32,35)(33,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,34)(27,38)(28,33)(30,40)(31,39)(32,37)(35,36);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,37)(26,35,33,38)(27,28,36,34)(29,32,39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=ghi,e^2=ghi,f^2=gh,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, 10080720);