#9240985

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

4245 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)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(2,6)(4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,34)(27,31,38,35)(28,32,29,33)(36,39,37,40);(2,6)(3,8)(10,14)(15,17)(16,20)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,30,38)(26,31,34,35)(28,36,29,37)(32,39,33,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 9148181);