Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #61630 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1186 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 148 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:
(1,2,6,3)(4,15,19,28)(5,9,21,11)(7,22,25,29)(8,12,24,14)(10,23,13,16)(17,30,20,31)(18,26,32,27);(2,9)(3,11)(5,21)(7,25)(10,15)(12,22)(13,28)(14,29)(16,23)(18,32)(26,30)(27,31);(1,3)(2,6)(4,13)(5,22)(7,9)(8,14)(10,19)(11,25)(12,24)(15,18)(16,30)(17,27)(20,26)(21,29)(23,31)(28,32);(1,4)(2,10)(3,13)(5,23)(6,19)(7,18)(8,20)(9,15)(11,28)(12,27)(14,26)(16,21)(17,24)(22,31)(25,32)(29,30)Polycyclic presentation:
2^9|a^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151696);