Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61661 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
881 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 881 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,7)(5,18)(6,11)(8,14)(9,16)(10,19)(13,27)(15,23)(17,25)(20,30)(21,32)(22,31)(24,28)(26,29);(2,7)(3,13)(4,16)(5,19)(8,22)(9,11)(10,25)(12,26)(14,28)(15,31)(18,21)(20,24)(23,27)(29,30);(1,3,4,15,6,14,16,30)(2,8,9,23,11,12,7,20)(5,13,17,29,19,28,32,31)(10,24,25,26,18,27,21,22);(2,7)(3,14)(4,16)(8,23)(9,11)(10,21)(12,20)(13,28)(17,32)(18,25)(22,27)(24,26)Polycyclic presentation:
2^9|c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407458);