Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3237 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)(13,14,18,17)(15,22,19,20)(16,24,25,28)(21,27,26,23);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,17,22)(16,23,25,27)(21,24,26,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,23)(17,26)(18,25)(19,27)(20,24)(22,28);(7,12)(9,11)(13,17,18,14)(15,22,19,20)(16,21,25,26)(23,24,27,28);(6,10)(9,11)(13,15,18,19)(14,22,17,20)(16,23,25,27)(21,28,26,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407441);