Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392135 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14679 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
15985 conjugacy classes of subgroups; only the 688 normal subgroups are shown.
This group has 15985 conjugacy classes of subgroups — too many to draw. Only its 688 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:
(1,2,5,10,6,8,18,22)(3,11,14,24,13,7,17,20)(4,9,16,23,15,21,28,31)(12,19,27,30,26,25,29,32);(1,3)(2,7)(4,12)(5,17)(6,13)(8,11)(9,19)(10,24)(14,18)(15,26)(16,29)(20,22)(21,25)(23,32)(27,28)(30,31);(2,8)(5,18)(7,11)(9,21)(14,17)(16,28)(19,25)(27,29);(1,4)(2,9)(3,12)(5,16)(6,15)(7,19)(8,21)(10,23)(11,25)(13,26)(14,27)(17,29)(18,28)(20,30)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,h^b=hi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72213500, #72213538, #72227588, … |
| maximal normal subgroup of | #72213500, #72213538, #72227588, … |
| maximal quotient of | #72233910, #72234502, #72235014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299332);