Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2382 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)(7,11)(8,10)(9,12)(14,19)(15,22)(16,18)(20,27)(23,24)(26,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14,17,19)(15,20,24,26)(16,25,18,21)(22,28,23,27);(2,4)(13,15,16,23,17,24,18,22)(14,20,25,27,19,26,21,28);(6,10)(9,11)(13,16,17,18)(14,21,19,25)(15,23,24,22)(20,28,26,27);(2,4)(7,12)(9,11)(14,19)(20,26)(21,25)(27,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307388, #72307519, #72307669, … |
| maximal normal subgroup of | #72307388, #72307519, #72307669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590198);