Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996315 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2293 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)(13,14,18,23,19,21,17,27)(15,24,16,22,25,28,26,20);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,17)(18,26)(19,25)(21,22)(23,28)(24,27);(2,4)(5,7)(6,9)(8,12)(10,11)(14,21)(15,25)(16,26)(23,27);(7,12)(9,11)(13,16,17,15,19,26,18,25)(14,22,27,24,21,20,23,28);(6,10)(9,11)(13,17,19,18)(14,23,21,27)(15,26,25,16)(20,24,22,28)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282978, #72283245, #72285334, … |
| maximal normal subgroup of | #72282978, #72283245, #72285334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913894);