#9515013

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

4530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=i,f^2=g,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, 9422209);