Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14885 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2580 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)(3,8)(4,15)(5,18)(6,11)(7,13)(9,17)(10,21)(12,24)(14,25)(16,26)(19,23)(20,27)(22,30)(28,31)(29,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,18)(9,22)(10,25)(11,23)(13,21)(15,31)(16,30)(17,29)(20,28)(24,26)(27,32);(1,3)(2,8)(4,12)(5,19)(6,14)(7,10)(9,24)(11,25)(13,21)(15,32)(16,30)(17,29)(18,23)(20,28)(22,27)(26,31);(1,4)(2,9,11,15)(3,12)(5,20)(6,17)(7,22,23,31)(8,24,25,32)(10,27,18,26)(13,30)(14,29)(16,21)(19,28)Polycyclic presentation:
2^9|d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261457, #72279915, #72286033, … |
| maximal normal subgroup of | #72261457, #72279915, #72286033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311193);