Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500268 |
| 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 | #61673 | 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 | |
853 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 853 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:
(2,7)(3,11)(4,16)(5,19)(8,23)(9,25)(10,17)(12,28)(13,21)(15,24)(20,22)(26,29);(1,2,6,10)(3,12,11,26)(4,17,16,7)(5,20,19,9)(8,24,13,15)(14,28,27,29)(18,22,31,25)(21,30,23,32);(1,3,4,14,6,11,16,27)(2,8,7,21,10,13,17,23)(5,15,18,30,19,24,31,32)(9,12,22,29,20,26,25,28);(1,2)(3,13)(4,17)(5,9)(6,10)(7,16)(8,11)(12,24)(14,21)(15,26)(18,25)(19,20)(22,31)(23,27)(28,32)(29,30)Polycyclic presentation:
2^9|b^2=i,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,d^b=di,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407465);