Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,10,7,9,11,8,12)(14,18,20,25)(15,21)(16,23)(19,24,27,26)(22,28);(2,4)(5,7,10,11,9,12,8,6)(14,19,20,27)(15,22)(18,24,25,26)(21,28);(5,8,9,10)(6,7,11,12)(13,14,15,18,16,19,21,26,17,20,22,25,23,27,28,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cgh,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318512, #72318549, #72318594, … |
| maximal normal subgroup of | #72318512, #72318549, #72318594, … |
| maximal quotient of | #72320860, #72322458, C2×#140369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47565);