Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(14,19)(16,17)(20,26)(21,25)(22,24)(27,28);(6,9)(7,11)(13,14)(15,20)(16,19)(17,25)(18,21)(22,26)(23,27)(24,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,22)(17,24)(18,23)(19,26)(21,27)(25,28);(6,9)(10,12)(13,16)(15,22)(17,18)(19,25)(23,24)(26,28);(5,7,8,11)(6,10,9,12)(15,23)(20,27)(22,24)(26,28)Polycyclic presentation:
2^9|c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280241, #72280269, #72281030, … |
| maximal normal subgroup of | #72280241, #72280269, #72281030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7062059);