Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1718 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(20,27)(22,28)(23,30)(25,31);(3,10)(7,14)(11,16)(13,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^9|d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306483, #72306487, #72306490, … |
| maximal normal subgroup of | #72306483, #72306487, #72306490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141513);