#9221212

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

4606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9128408);