Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,24)(7,12,19,27,17,26,18,33)(8,20,23,28,21,34,22,13)(16,29,31,35,32,36,30,25);(1,2)(3,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,25)(14,27)(15,26)(20,29)(21,32)(22,30)(23,31)(24,33)(28,35)(34,36);(5,8,9,22,11,21,10,23)(6,13,14,20,15,28,24,34)(7,16,18,30,17,32,19,31)(12,25,27,29,26,35,33,36);(2,4);(7,17)(8,21)(12,26)(13,28)(18,19)(20,34)(22,23)(27,33)Polycyclic presentation:
2^9|a^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72343918 |
| maximal normal subgroup of | #72343918 |
| maximal quotient of | #72343959 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287297);