Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2854 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,24)(17,20)(18,21)(22,26)(23,28)(25,27);(5,7)(6,9)(8,12)(10,11)(15,22)(16,17)(19,26)(20,24)(23,25)(27,28);(2,4)(13,15,18,23)(14,19,21,28)(16,22,17,25)(20,27,24,26);(2,4)(6,10)(7,12)(13,14)(15,19)(16,24)(17,20)(18,21)(22,26)(23,28)(25,27);(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,27,28,26)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307372, #72307705, #72307847, … |
| maximal normal subgroup of | #72307372, #72307705, #72307847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410361);