Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508062 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1120 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18,21,24)(19,25,27,32)(20,23,28,31)(22,29,30,26);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(20,28)(23,31);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32);(4,12)(7,14)(10,16)(13,15)Polycyclic presentation:
2^9|a^2=ei,b^2=e,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236618, #72236762, #72236767, … |
| maximal normal subgroup of | #72236618, #72236762, #72236767, … |
| maximal quotient of | #72279013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415333);