Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3678 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)(7,11)(8,10)(9,12)(13,14,17,22,18,23,16,21)(15,19,24,27,25,28,26,20);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,26)(17,24)(18,25)(20,21)(22,27)(23,28);(1,2,3,4)(13,15)(14,20)(16,24)(17,26)(18,25)(19,21)(22,28)(23,27);(7,12)(9,11);(6,10)(9,11)(13,16,18,17)(14,21,23,22)(15,24,25,26)(19,27,28,20)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72292850, #72292856, #72292860 |
| maximal normal subgroup of | #72292850, #72292856, #72292860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300607);