Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 | |
2484 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)(13,14,18,21)(15,22,25,27)(16,20,17,26)(19,23,28,24);(2,3)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,15,18,25)(14,19,21,28)(16,24,17,23)(20,27,26,22);(1,3,4,2)(7,12)(9,11)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,28,27);(6,10)(9,11)(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,27,28,22)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281225, #72281862 |
| maximal normal subgroup of | #72281225, #72281862 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903634);