Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #24218 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 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)(8,12)(9,10)(14,20)(15,24)(16,27)(17,19)(21,25)(22,26);(1,2)(3,4)(6,10)(11,12)(13,14,18,20)(15,25,26,16)(17,23,19,28)(21,24,27,22);(5,7,9,8)(6,11,10,12)(13,15,19,24,18,26,17,22)(14,21,23,16,20,27,28,25);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,21,18,25,19,27)(14,22,28,26,20,24,23,15);(5,8,9,7)(6,11,10,12)(13,17,18,19)(14,23,20,28)(15,22,26,24)(16,27,25,21)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282233, #72308239, #72308696, … |
| maximal normal subgroup of | #72282233, #72308239, #72308696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472420);