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Cayley diagram

C2×#135 #572

order64
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = #72214076 (order 1024) · Inn(G) = D4 · Out(G) = #2403
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#135

Subgroups

subgroupG/H
centerC2×C4D4
commutator, Frattini, cosocleC4C24
socleC22C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1876, #1974, #1975, …
maximal normal subgroup of#1876, #1974, #1975, …
maximal quotient ofC2×#416, C2×#427, C2×C4×D8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 253);