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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2508 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)(13,14,18,17)(15,19,25,24)(16,20,26,21)(22,27,28,23);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,24)(18,25)(20,22)(21,28)(26,27);(2,4)(5,8,9,10)(6,7,11,12)(13,16)(14,20)(15,22)(17,21)(18,26)(19,27)(23,24)(25,28);(2,4)(6,11)(7,12)(13,16)(14,21)(15,23)(17,20)(18,26)(19,28)(22,24)(25,27);(13,14,18,17)(15,24,25,19)(16,20,26,21)(22,23,28,27)Polycyclic presentation:
2^9|a^2=i,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, #72307760, #72307866, … |
| maximal normal subgroup of | #72307408, #72307760, #72307866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408416);