#9161007

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

4347 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=gi,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, 9068203);