Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 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)(15,19)(16,18)(17,20)(21,24)(22,26)(23,25)(27,28);(2,3)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(1,3,4,2)(15,22)(19,26)(23,28)(25,27);(5,7,8,11)(6,10,9,12)(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256846, #72256864, #72256885, … |
| maximal normal subgroup of | #72256846, #72256864, #72256885, … |
| maximal quotient of | #72265238, C2×#6467152, #72311342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374348);