Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | |
2290 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)(13,14,18,22)(15,19,23,27)(16,21,17,20)(24,26,25,28);(1,2)(3,4)(6,9)(7,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,28)(21,26)(22,27);(5,6)(7,10)(8,9)(11,12)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(2,4)(6,9)(10,12)(13,16,18,17)(14,21,22,20)(15,24,23,25)(19,26,27,28);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72257358, #72257494, #72264140, … |
| maximal normal subgroup of | #72257358, #72257494, #72264140, … |
| maximal quotient of | #72265665 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050042);