#9794224

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 9701420);