#9128639

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	#44001	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9035835);