#9250321

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

4667 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,e^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, 9157517);