Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1262 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,13)(4,8)(5,22)(6,11)(7,12)(9,14)(10,18)(15,29)(16,31)(17,26)(19,23)(20,25)(21,27)(24,28)(30,32);(2,8)(4,18)(5,23)(9,13)(10,11)(12,22)(14,20)(16,28)(19,27)(24,29)(25,26)(31,32);(1,2)(3,9)(4,19)(5,10)(6,11)(7,12)(8,23)(13,28)(14,29)(15,24)(16,26)(17,25)(18,22)(20,32)(21,27)(30,31);(1,3)(2,9)(4,20)(5,16)(6,17)(7,15)(8,13)(10,26)(11,25)(12,24)(14,18)(19,32)(21,30)(22,29)(23,28)(27,31)Polycyclic presentation:
2^9|e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267805 |
| maximal normal subgroup of | #72267805 |
| maximal quotient of | #72267810 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151695);