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Cayley diagram

C2×#12722 #493448

order512
nilpotency class4
isoclinismisoclinic class of #12721
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#12722

Subgroups

subgroupG/H
center, socleC22#1782
commutatorC22×C4C23×C4
Frattini, cosocleC2×#41C24
prime radicalC2×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

2570 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)(5,12,14,8)(6,10)(7,11)(9,13,17,18)(15,16);(4,8)(6,15)(9,16)(10,17)(11,12)(13,18);(3,5,7,14)(4,9)(6,13,15,18)(8,16)(10,12)(11,17)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72233316, #72260388, C2×#511846, …
maximal normal subgroup of#72233316, #72260388, C2×#511846, …
maximal quotient of#72233309, C2×#147734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400644);