Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | |
3022 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)(18,20)(21,26)(24,27)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(20,22)(21,27)(23,28);(2,4)(14,20)(16,18)(19,22)(21,28)(23,27)(24,26);(6,10)(7,12)(13,15)(14,21)(16,24)(17,25)(18,26)(19,27)(20,28)(22,23);(13,16,17,18)(14,22,19,20)(15,24,25,26)(21,23,27,28)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,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249598, #72249639 |
| maximal normal subgroup of | #72249595, #72249598, #72249639 |
| maximal quotient of | #72264490, #72264692, #72264762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407895);