Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
894 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 894 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,29,30,26)(24,28)(25,27)(31,35,36,33)(32,34);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23,24,31,25,30,32,36)(22,26,28,33,27,29,34,35);(1,2,3,4)(7,14)(10,18)(13,19)(15,20)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240257, #72240340, #72240343, … |
| maximal normal subgroup of | #72240257, #72240340, #72240343, … |
| maximal quotient of | #72257737, #72286532 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18300);