Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,24,16,20)(17,23,19,27)(21,25,28,26);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,18)(17,26)(19,25)(21,27)(22,24)(23,28);(1,3,4,2)(7,11)(9,12)(13,16,18,15)(14,21,22,28)(17,26,19,25)(20,23,24,27);(6,10)(9,12)(14,22)(20,24)(21,28)(23,27);(13,17,18,19)(14,23,22,27)(15,25,16,26)(20,21,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314893, #72314898, #72314933, … |
| maximal normal subgroup of | #72314893, #72314898, #72314933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340408);