#9863226

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

4565 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,18,20,15)(12,21,22,17)(13,23,24,19)(25,26)(27,32)(28,34)(29,35)(30,40)(31,33)(36,37)(38,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,19)(13,17)(14,20)(16,18)(21,24)(22,23)(25,27,31,30)(26,32,33,40)(28,36,35,38)(29,39,34,37);(1,2)(3,5)(4,6)(7,8)(10,16)(12,22)(15,18)(19,23)(26,33)(28,35)(32,40)(36,38);(1,3,4,8)(2,5,6,7)(9,12,14,22)(10,17,16,21)(11,19,20,23)(13,18,24,15)(25,28,31,35)(26,34,33,29)(27,36,30,38)(32,37,40,39);(9,11)(10,18)(12,19)(13,21)(14,20)(15,16)(17,24)(22,23)(25,29,31,34)(26,35,33,28)(27,37,30,39)(32,36,40,38);(9,13,14,24)(10,19,16,23)(11,17,20,21)(12,18,22,15)(25,30,31,27)(26,32,33,40)(28,38,35,36)(29,39,34,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770422);