Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1420 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1420 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,22)(10,14)(11,18)(13,23)(16,30)(17,31)(19,28)(20,27)(21,29)(24,35)(25,34)(26,32)(33,36);(5,7)(6,11,14,25)(8,16)(9,18)(10,17)(12,26,29,24)(13,15,22,31)(19,35)(20,32)(21,33)(23,34)(27,36,28,30);(2,4)(5,8)(6,12)(7,16)(9,19)(10,21)(11,24)(13,27)(14,29)(15,30)(17,33)(18,35)(20,23)(22,28)(25,26)(31,36)(32,34);(7,17)(8,19,21,20)(11,25)(12,27,29,28)(15,31)(16,32,33,35)(18,34)(24,36,26,30)Polycyclic presentation:
2^9|b^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213162, C2×#475763, #72216224, … |
| maximal normal subgroup of | #72213162, C2×#475763, #72216224, … |
| maximal quotient of | #72213145, #72213241, #72213348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382959);