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 | C2×#3179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14,18,20)(15,23,27,16)(17,28,24,22)(19,21,25,26);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,24,21,18,27,17,26)(14,19,28,16,20,25,22,23);(5,8,9,10)(6,7,11,12)(13,14)(15,25)(16,26)(17,22)(18,20)(19,27)(21,23)(24,28);(2,4)(6,11)(7,12)(14,20)(16,23)(19,25)(22,28);(13,16,17,19,18,23,24,25)(14,21,22,15,20,26,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257092, #72257122, #72257158 |
| maximal normal subgroup of | #72257092, #72257122, #72257158 |
| maximal quotient of | #72308715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449228);