#9206417

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
prime radical	#47836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113613);