Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129559 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,10,14,26,16)(7,17,19,31,21,12,20,30)(8,22,24,13,25,11,23,28)(18,27,34,29,36,32,35,33);(1,2)(3,4)(6,11)(8,23)(9,26)(12,27)(13,15)(14,22)(16,28)(17,32)(18,34)(19,20)(24,25)(29,31)(30,33)(35,36);(2,4)(5,7)(6,12)(8,18)(9,20)(10,21)(11,27)(13,29)(14,17)(15,31)(16,30)(19,26)(22,32)(23,34)(24,35)(25,36)(28,33)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249868, #72249878, #72249897, … |
| maximal normal subgroup of | #72249868, #72249878, #72249897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54830);