Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,19,24,28)(16,21,17,20)(23,27,25,26);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(5,7,8,12)(6,9,10,11)(14,20)(16,17)(19,26)(21,22)(23,25)(27,28);(1,3,4,2)(7,12)(9,11)(13,16,18,17)(14,21,22,20)(15,23,24,25)(19,27,28,26);(6,10)(9,11)(14,22)(15,24)(20,21)(23,25)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963025);