Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4296 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,23,24,19)(16,21,18,20)(25,27,26,28);(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,27)(21,28)(22,23);(14,20)(16,18)(19,27)(21,22)(23,28)(25,26);(2,4)(15,24)(19,23)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,20)(15,25,24,26)(19,28,23,27)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257467, #72257512, #72263990, … |
| maximal normal subgroup of | #72257467, #72257512, #72263990, … |
| maximal quotient of | #72308336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536529);