Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| 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 | |
563 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 108 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)(7,9)(8,11)(10,12)(13,14,15,19,23,27,28,26,17,21,24,25,16,20,22,18);(1,2,3,4)(14,18)(15,22)(16,23)(19,20)(21,26)(24,28)(25,27);(6,9)(7,12)(10,11)(14,18)(15,22)(16,23)(19,20)(21,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=ghi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225208, #72226848, #72226892 |
| maximal normal subgroup of | #72225208, #72226848, #72226892 |
| maximal quotient of | #72225198, #72226844, #72226847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44732);