Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420932 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15282 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,7,12,6,11)(3,13,16,25,14,23,21,8)(4,17,19,26,20,10,22,28)(15,29,18,24,30,27,31,32);(1,3,6,16,7,14,5,21)(2,8,11,13,12,25,9,23)(4,18,19,15,20,31,22,30)(10,27,28,24,17,29,26,32);(2,9)(3,14)(5,6)(8,23)(10,26)(11,12)(13,25)(15,30)(17,28)(19,22)(24,27)(29,32);(1,4)(2,10)(3,15)(5,22)(6,19)(7,20)(8,24)(9,26)(11,28)(12,17)(13,29)(14,30)(16,31)(18,21)(23,27)(25,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251787, #72251792, #72251795, … |
| maximal normal subgroup of | #72251787, #72251792, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328128);