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Cayley diagram

C8×Q32 #12042

order256
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C23×#495725 (order 4096) · Inn(G) = D8 · Out(G) = C23×C4×D4
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

direct productC8×Q32

Subgroups

subgroupG/H
centerC2×C8D8
commutatorC8C22×C8
Frattini, cosocleC4×C8C23
socle, prime radicalC22C4×D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C82 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#93748, #93750, C16×Q32, …
maximal normal subgroup of#93748, #93750, C16×Q32, …
maximal quotient ofC16×Q32, #122949, #122956, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5030);