#9243475

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

4195 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150671);