Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #37658 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2302 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)(7,10)(8,11)(9,12)(14,19)(17,20)(21,26)(22,27)(23,28)(24,25);(5,7)(6,9)(8,12)(10,11)(13,14,20,27,18,22,17,19)(15,23,25,26,16,21,24,28);(2,4)(7,12)(9,10)(13,15)(14,21)(16,18)(17,24)(19,26)(20,25)(22,23)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,15)(14,21,22,23)(17,25,20,24)(19,26,27,28);(7,12)(9,10)(14,22)(15,16)(19,27)(24,25)Polycyclic presentation:
2^9|b^2=fh,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264011, #72264050, #72264074 |
| maximal normal subgroup of | #72264011, #72264050, #72264074 |
| maximal quotient of | #72264952 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504322);