Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(5,7)(6,9)(8,11)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(1,2,3,4)(6,10)(7,11)(13,15)(14,20)(16,23)(17,22)(18,24)(19,26)(21,27)(25,28);(7,11)(9,12);(15,22)(20,27)(23,26)(24,28)Polycyclic presentation:
2^9|b^2=fi,c^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281055, #72281863, #72310659, … |
| maximal normal subgroup of | #72281055, #72281863, #72310659, … |
| maximal quotient of | #72313063, #72313075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366855);