Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | #61431 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,21,27,23,25,22,37)(19,28,30,45,31,44,38,43)(20,32,34,42,36,47,35,26)(24,33,40,46,39,48,41,29);(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,39)(26,43)(27,40)(28,32)(29,35)(34,46)(36,48)(37,41)(42,45)(44,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,25)(24,39)(26,42)(27,37)(28,44)(32,47)(40,41)(43,45);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72320258, #72321764, #72321870 |
| maximal normal subgroup of | #72320258, #72321764, #72321870 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388605);