Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(2,4)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(6,9)(10,12)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(15,22)(18,26)(21,23)(25,27);(14,20)(18,26)(19,28)(25,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|c^2=i,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72257288, #72257308, #72257361, … |
| maximal normal subgroup of | #72257288, #72257308, #72257361, … |
| maximal quotient of | #72264483, #72310221, C4×#35000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535558);