Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,17,21,18)(16,24,25,20)(23,27,28,26);(1,2)(3,4)(13,15,18,14,19,21,17,22)(16,23,27,20,25,28,26,24);(2,4)(6,9)(10,12)(13,16)(14,20)(15,23)(17,26)(18,27)(19,25)(21,28)(22,24);(2,4)(16,25)(20,24)(23,28)(26,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,15)(16,26,25,27)(20,28,24,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282046, #72282224 |
| maximal normal subgroup of | #72282046, #72282224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361770);