Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16245 |
| 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 | |
679 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,16)(5,13)(9,21)(12,25)(14,26)(22,32);(1,2,5,18,6,7,13,10)(3,12,15,8,11,25,27,20)(4,9,17,24,16,21,19,23)(14,29,30,22,26,28,31,32);(1,3,13,27,6,11,5,15)(2,8,18,25,7,20,10,12)(4,14,17,30,16,26,19,31)(9,22,23,29,21,32,24,28);(1,4)(2,9,7,21)(3,14)(5,19)(6,16)(8,22,20,32)(10,24,18,23)(11,26)(12,28,25,29)(13,17)(15,31)(27,30)Polycyclic presentation:
2^9|b^2=efg,c^2=egh,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338910, #72338912, #72338927 |
| maximal normal subgroup of | #72338910, #72338912, #72338927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361136);