Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14795 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1419 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)(3,11)(4,15)(5,18)(6,10)(7,13)(8,17)(9,20)(12,26)(14,24)(16,23)(19,21)(22,31)(25,28)(27,29)(30,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,22)(9,24)(10,21)(11,18)(13,20)(15,29)(16,31)(17,28)(23,26)(25,30)(27,32);(7,21)(11,24)(22,29)(25,26);(1,4,5,16,6,17,20,32)(2,8,9,23,10,15,18,30)(3,12,19,31,14,28,13,27)(7,22,24,26,21,29,11,25)Polycyclic presentation:
2^9|d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216059, #72216142, … |
| maximal normal subgroup of | #72216013, #72216059, #72216142, … |
| maximal quotient of | C2×#400046, #72252109, #72252208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307242);