Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | |
2400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,28,21)(16,18,23,20)(17,26,27,22);(2,3)(6,9)(7,11)(13,15)(14,21)(16,26)(17,18)(19,28)(20,27)(22,23)(24,25);(1,2,4,3)(13,16,20,14,19,23,18,24)(15,26,27,21,28,22,17,25);(6,9)(10,12)(13,17,19,27)(14,22,24,26)(15,18,28,20)(16,21,23,25);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,16)(15,27,28,17)(21,26,25,22)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723270);