Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(11,24)(15,28)(16,30)(31,34);(5,6)(7,15)(8,12)(9,21)(10,13)(11,18)(14,23)(16,29)(17,28)(19,27)(20,33)(22,24)(25,31)(26,32)(30,36)(34,35);(5,7)(6,11)(8,16)(9,22)(10,17)(12,25)(13,24)(14,28)(15,21)(18,23)(19,30)(20,34)(26,36)(27,35)(29,33)(31,32);(5,8,10,19)(6,12,13,27)(7,16,17,30)(9,20,23,32)(11,25,24,35)(14,26,21,33)(15,29,28,36)(18,31,22,34)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265644, #72265649, #72265671, … |
| maximal normal subgroup of | #72265644, #72265649, #72265671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266773);