Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,21)(9,16)(10,14)(11,15)(12,19)(13,23)(18,32)(20,30)(22,29)(24,28)(25,26)(27,34)(31,33)(35,36);(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,27)(14,30)(15,26)(16,17)(21,31)(22,35)(23,33)(24,34)(28,36)(29,32);(12,26)(17,30)(27,32)(31,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,28,15,29,16,21)(7,18,25,35,20,34,19,33)(12,27,30,36,26,32,17,31)Polycyclic presentation:
2^9|d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256515, #72256516, #72256522 |
| maximal normal subgroup of | #72256515, #72256516, #72256522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307243);