Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2140 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)(7,9)(8,10)(11,12)(13,14,18,17,19,21,27,28)(15,22,24,23,20,25,26,16);(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,25)(18,26)(19,20)(21,23)(22,28)(24,27);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,21)(17,26)(18,22)(19,23)(24,28)(25,27);(7,12)(9,11)(13,17,27,14,19,28,18,21)(15,23,26,22,20,16,24,25);(6,10)(9,11)(15,20)(16,23)(22,25)(24,26)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307916, #72308047, #72308306, … |
| maximal normal subgroup of | #72307916, #72308047, #72308306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972097);