#10086347

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	#46379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993543);