Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1107 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1107 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,30)(17,31)(19,26)(20,35)(21,28)(22,36)(23,29)(24,32)(25,33)(27,34);(1,2,3,4)(7,16)(8,19)(11,24)(12,26)(15,30)(18,32)(21,22)(28,36);(5,7,10,16)(6,11,14,24)(8,20,19,17)(9,18,23,32)(12,27,26,25)(13,15,29,30)(21,34,22,33)(28,35,36,31);(5,8,10,19)(6,12,14,26)(7,17,16,20)(9,22,23,21)(11,25,24,27)(13,28,29,36)(15,31,30,35)(18,34,32,33)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,d^c=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72312445, #72312448, #72312456, … |
| maximal normal subgroup of | #72312445, #72312448, #72312456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419115);