Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | #37650 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2969 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)(15,21)(16,23)(18,25)(20,27)(22,24)(26,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,19)(15,18,22,26)(16,20,23,27)(21,25,24,28);(2,4)(13,15,17,22)(14,18,19,26)(16,24,23,21)(20,28,27,25);(6,10)(9,11)(14,19)(18,26)(20,27)(25,28);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|b^2=i,c^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=efh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72239649, #72282743, #72283136, … |
| maximal normal subgroup of | #72239649, #72282743, #72283136, … |
| maximal quotient of | #72239772, #72239773, #72239775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575188);