Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352157 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,5,8,7)(10,14)(11,17)(12,19)(15,22)(16,20)(21,23);(2,5)(3,8)(6,7)(9,10)(11,15)(12,20)(13,16)(14,19)(17,23)(18,22)(21,24);(12,19)(14,20)(18,24)(21,22);(2,6)(5,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,23)(19,24)(20,22);(11,17)(15,23)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281484, #72281576, #72281866 |
| maximal normal subgroup of | #72281484, #72281576, #72281866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259491);