Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2733 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16609 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
754 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)(3,13)(4,18)(5,8)(6,11)(7,12)(9,16)(10,20)(14,30)(15,23)(17,25)(19,29)(21,24)(22,28)(26,32)(27,31);(2,8,12,11)(9,25,13,23)(10,24,28,18)(26,30,29,27);(1,3)(2,9,12,13)(4,19)(5,17)(6,15)(7,16)(8,23,11,25)(10,27,28,30)(14,20)(18,26,24,29)(21,32)(22,31);(1,4)(2,10)(3,14,16,31)(5,20)(6,22)(7,21)(8,24)(9,26,13,29)(11,18)(12,28)(15,19,17,32)(23,27,25,30)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279719 |
| maximal normal subgroup of | #72279719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375589);