Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14796 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,20)(3,12,14,22,15,24,19,8)(4,16,17,9,18,25,21,26)(13,28,29,32,30,23,31,27);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,23)(10,24)(11,22)(12,20)(16,27)(17,31)(18,30)(21,29)(25,32)(26,28);(8,22)(12,24)(23,28)(27,32);(1,4,5,17,7,18,6,21)(2,9,10,25,11,26,20,16)(3,13,19,31,15,30,14,29)(8,23,24,32,22,28,12,27)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,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×#400131, #72252113, #72256341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307327);