#9290482

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

4394 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=gi,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, 9197678);