Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2489 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)(13,14,17,19)(15,22,21,18)(16,24,26,20)(23,27,25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,21)(20,24)(23,25);(2,4)(13,15,18,14,17,21,22,19)(16,25,27,24,26,23,28,20);(6,10)(9,11)(13,16,17,26)(14,20,19,24)(15,23,21,25)(18,27,22,28);(2,4)(7,12)(9,11)(16,26)(20,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283039, #72283104, #72283188, … |
| maximal normal subgroup of | #72283039, #72283104, #72283188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589643);