Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2205 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)(13,14)(15,19)(16,21)(17,20)(18,22)(23,26)(24,27)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,22,20)(16,24,25,23)(21,27,28,26);(6,9)(10,12)(14,20)(15,17)(19,22)(21,27)(23,24)(26,28);(6,9)(10,12)(13,16)(14,21)(15,23)(17,24)(18,25)(19,26)(20,27)(22,28);(5,7,8,11)(6,10,9,12)(14,22)(16,25)(19,20)(23,24)Polycyclic presentation:
2^9|b^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308121 |
| maximal normal subgroup of | #72308121 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6399028);