Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3080 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)(7,9)(8,10)(11,12)(14,20)(21,27)(22,28)(23,24);(2,4)(5,7,8,11)(6,9,10,12)(13,14)(15,22)(16,27)(17,24)(18,20)(19,23)(21,25)(26,28);(6,10)(9,12)(13,15,18,26)(14,21,20,27)(16,17,25,19)(22,24,28,23);(13,16)(14,22)(15,17)(18,25)(19,26)(20,28)(21,24)(23,27);(13,17,18,19)(14,23,20,24)(15,25,26,16)(21,22,27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281881, #72281966, #72282379, … |
| maximal normal subgroup of | #72281881, #72281966, #72282379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347710);