#9794260

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

4295 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=h,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, 9701456);