Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499761 |
| 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×#3186 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,7,9)(8,10,12,11)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(1,2,3,4)(6,9)(10,11)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(6,10)(8,12)(9,11)(13,16)(14,19)(15,21)(17,24)(18,25)(20,28)(22,23)(26,27);(16,24)(19,28)(21,23)(25,27);(15,22)(18,26)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72283392, #72283638, #72314244, … |
| maximal normal subgroup of | #72283392, #72283638, #72314244, … |
| maximal quotient of | #72314391 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415908);