Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
5044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,19)(17,22)(18,23)(20,24)(25,28)(26,27);(2,4)(5,7)(6,9)(8,11)(10,12)(14,19)(16,17)(20,27)(21,28)(22,23)(24,25);(7,11)(9,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,22)(23,28);(7,11)(9,12)(13,15)(14,21)(16,25)(17,24)(18,26)(19,28)(20,22)(23,27);(6,10)(9,12)(13,16,18,17)(14,19,23,22)(15,24,26,25)(20,27,28,21)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257071, #72257082, #72257133, … |
| maximal normal subgroup of | #72257071, #72257082, #72257133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377409);