Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16717 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,19,6,10,5,11)(3,12,14,22,15,27,20,8)(4,9,17,23,18,24,16,25)(13,28,29,21,31,26,30,32);(1,3)(2,8,10,22)(4,13)(5,14)(6,15)(7,20)(9,21,24,32)(11,12,19,27)(16,29)(17,30)(18,31)(23,28,25,26);(1,4)(2,9)(3,13,15,31)(5,16)(6,18)(7,17)(8,21,22,32)(10,24)(11,25)(12,26,27,28)(14,29,20,30)(19,23);(2,10)(4,16)(5,7)(8,22)(9,23)(13,29)(14,20)(17,18)(21,28)(24,25)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251693, #72251706, … |
| maximal normal subgroup of | #72251685, #72251693, #72251706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378123);