Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35514 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(14,19)(20,25)(21,26)(27,28);(5,7)(6,10)(8,11)(9,12)(14,20)(16,18)(19,25)(21,27)(22,24)(26,28);(2,4)(5,8,9,10)(6,7,11,12)(13,14)(15,21)(16,20)(17,19)(18,25)(22,28)(23,26)(24,27);(2,4)(6,11)(7,12)(13,15)(14,21)(16,24)(17,23)(18,22)(19,26)(20,27)(25,28);(13,16,17,18)(14,20,19,25)(15,22,23,24)(21,28,26,27)Polycyclic presentation:
2^9|c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307408, #72307781, #72307847 |
| maximal normal subgroup of | #72307408, #72307781, #72307847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408415);