Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,22)(17,19)(18,26)(20,24)(21,27)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(20,21)(22,26)(23,28);(2,4)(13,15)(14,20)(16,24)(17,25)(18,27)(19,21)(22,28)(23,26);(6,10)(7,12)(13,15)(14,21)(16,24)(17,25)(18,27)(19,20)(22,23)(26,28);(13,16,17,18)(14,22,19,26)(15,24,25,27)(20,28,21,23)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237210, #72249595, #72249598, … |
| maximal normal subgroup of | #72237210, #72249595, #72249598, … |
| maximal quotient of | #72238931, #72264579, #72264762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407856);