Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3799 conjugacy classes of subgroups; only the 418 normal subgroups are shown.
This group has 3799 conjugacy classes of subgroups — too many to draw. Only its 418 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,15,21,11)(7,12,17,25,18,26,32,22)(8,13,19,27,20,28,33,23)(16,29,30,34,31,24,35,36);(1,2,3,4)(6,11)(9,21)(12,22)(13,23)(14,15)(17,32)(19,33)(24,34)(25,26)(27,28)(29,36)(30,35);(5,7)(6,12)(8,16)(9,17)(10,18)(11,22)(13,24)(14,25)(15,26)(19,30)(20,31)(21,32)(23,34)(27,36)(28,29)(33,35);(5,8)(6,13)(7,16)(9,19)(10,20)(11,23)(12,24)(14,27)(15,28)(17,30)(18,31)(21,33)(22,34)(25,36)(26,29)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72341982, #72342056, #72342074 |
| maximal subgroup of | #72281084, #72281191, #72281366, … |
| maximal normal subgroup of | #72281084, #72281191, #72281366, … |
| maximal quotient of | #72281568 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299904);