Finite Group Databasebeta

← Prev (#126557) Next (#126559) →

Cayley diagram

C2×#7141 #126558

order512
nilpotency class3
isoclinismisoclinic class of D8
automorphism?
homologyAbelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×#7141

Subgroups

subgroupG/H
centerC2×C4×C8D4
commutatorC4C2×C82
Frattini, cosocleC2×C4×C8C23
socleC23#336
prime radicalC24#97
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(11,12,15,20,16,21,30,38,17,22,31,39,14,19,28,36)(13,23,25,40,24,34,41,37,26,35,29,42,27,18,32,33);(3,4,5,7,6,8,9,10)(11,13)(12,18)(14,27)(15,29)(16,24)(17,26)(19,35)(20,37)(21,23)(22,34)(25,31)(28,41)(30,32)(33,39)(36,40)(38,42)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#129044, C2×#128250, C2×C8×#350
maximal normal subgroup ofC2×#129044, C2×#128250, C2×C8×#350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33754);