#9794230

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

4291 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)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,38)(28,33)(29,39)(30,35)(31,36)(32,37)(34,40);(1,2,6,4)(3,8,5,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24)(25,27)(26,28)(29,35)(30,34)(31,40)(32,33)(36,39)(37,38);(1,3,6,5)(2,7,4,8)(10,15)(14,17)(16,23)(22,24)(25,28,32,38)(26,33,37,27)(29,30,40,36)(31,39,35,34);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,29,32,40)(26,34,37,39)(27,35,33,31)(28,30,38,36);(1,4,6,2)(3,8,5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,30)(26,35)(27,39)(28,40)(29,38)(31,37)(32,36)(33,34);(1,5,6,3)(2,7,4,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,31,32,35)(26,36,37,30)(27,40,33,29)(28,39,38,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=gh,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, 9701426);