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×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12);(2,3)(5,7,8,11)(6,9,10,12)(13,14,19,24)(15,21,26,28)(16,27,25,20)(17,23,18,22);(1,2,4,3)(6,10)(9,12)(13,15,19,26)(14,20,24,27)(16,17,25,18)(21,23,28,22);(1,3,4,2)(13,16)(14,21)(15,17)(18,26)(19,25)(20,23)(22,27)(24,28);(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fi,d^2=f,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 | #72307335, #72307489 |
| maximal normal subgroup of | #72307335, #72307489 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348838);