Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16208 |
| 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)(3,12)(4,16)(5,8)(10,22)(13,25)(14,27)(23,32);(1,2,8,11,6,7,5,18)(3,13,15,9,12,25,19,21)(4,10,20,24,16,22,17,26)(14,29,30,23,27,28,31,32);(1,3,5,19,6,12,8,15)(2,9,18,13,7,21,11,25)(4,14,20,30,16,27,17,31)(10,23,24,28,22,32,26,29);(1,4)(2,10,7,22)(3,14)(5,20)(6,16)(8,17)(9,23,21,32)(11,26,18,24)(12,27)(13,28,25,29)(15,31)(19,30)Polycyclic presentation:
2^9|b^2=gh,c^2=eg,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,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339020, #72339022, #72339024, … |
| maximal normal subgroup of | #72339020, #72339022, #72339024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361094);