Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500250 |
| 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 | #61633 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 845 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:
(2,8)(3,11)(4,15)(5,17)(9,23)(10,22)(12,28)(13,30)(14,19)(16,32)(18,26)(21,24)(25,29)(27,31);(1,2,7,4)(3,12,14,9)(5,15,17,8)(6,16,20,10)(11,25,19,24)(13,21,31,29)(18,22,26,32)(23,30,28,27);(1,3,5,13,7,14,17,31)(2,9,8,21,4,12,15,29)(6,19,18,30,20,11,26,27)(10,25,22,28,16,24,32,23);(1,4,7,2)(3,12,14,9)(5,8,17,15)(6,16,20,10)(11,25,19,24)(13,21,31,29)(18,22,26,32)(23,30,28,27)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,e^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,d^c=dh,f^c=fi,g^c=gi,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, 407450);