Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| prime radical | #24444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,23)(15,16,26,20)(17,24,19,22)(21,28,25,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,23)(17,21)(18,26)(19,25)(22,28)(24,27);(13,16)(14,21)(15,24)(17,27)(18,20)(19,28)(22,26)(23,25);(2,4)(5,8,9,7)(6,10,11,12)(14,22)(15,25)(17,19)(21,26)(23,24)(27,28);(6,11)(10,12)(13,17,18,19)(14,22,23,24)(15,21,26,25)(16,27,20,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282481 |
| maximal normal subgroup of | #72282481 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424907);