Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,7,10,8,11,12,9)(14,19)(15,22)(16,20)(17,25)(21,28)(23,26)(24,27);(1,2,3,4)(6,9)(7,12)(10,11)(13,14,20,27,17,21,26,22,18,15,23,28,25,24,16,19);(13,15,23,27,17,24,16,22,18,14,20,28,25,21,26,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=egi,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226848, #72226890, #72226895 |
| maximal normal subgroup of | #72226848, #72226890, #72226895 |
| maximal quotient of | #72226845, #72226853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48454);