Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,10,16)(7,17,21,32)(8,14,24,31)(9,25,11,15)(12,18,29,19)(13,26,30,23)(20,33,22,27)(28,35,36,34);(5,7,11,22,10,21,9,20)(6,12,15,28,16,29,25,36)(8,18,23,34,24,19,26,35)(13,27,31,17,30,33,14,32);(1,3,4,2)(5,8)(6,13)(7,18)(9,26)(10,24)(11,23)(12,27)(14,25)(15,31)(16,30)(17,28)(19,21)(20,35)(22,34)(29,33)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gh,d^b=dh,h^b=hi,e^c=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311927, #72311946, #72311964, … |
| maximal normal subgroup of | #72311927, #72311946, #72311964, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54522);