#9712035

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

4243 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619231);