Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(5,19)(8,21)(10,20)(12,28)(16,26)(17,18)(23,24)(27,30)(29,31);(1,2)(3,8)(4,16)(5,20)(6,10)(7,19)(9,17)(11,24)(12,29)(13,21)(14,23)(15,25)(18,26)(22,30)(27,31)(28,32);(1,3)(2,8)(4,12)(5,14)(6,11)(7,21)(9,22)(10,24)(13,19)(15,30)(16,31)(17,28)(18,27)(20,23)(25,32)(26,29);(1,4,6,18)(2,9)(3,12,11,27)(5,17,19,15)(7,16)(8,22)(10,25)(13,30,14,28)(20,26)(21,31)(23,29)(24,32)Polycyclic presentation:
2^9|d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338544, #72338552, #72338561 |
| maximal normal subgroup of | #72338544, #72338552, #72338561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334503);