#9787330

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

4521 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,f^2=g,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, 9694526);