#9492262

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9399458);