Finite Group Databasebeta

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Cayley diagram

C2×#12012 #483343

order512
nilpotency class3
isoclinismisoclinic class of D8
automorphism?
homologyAbelianization (H1) = C23×C16 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#12012

Subgroups

subgroupG/H
centerC2×C4×C8D4
commutatorC4C23×C16
Frattini, cosocleC4×C8C24
socleC23C2×C4⋊C8
prime radicalC24C4⋊C8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#122919, #72231978, #72231980
maximal normal subgroup ofC2×#122919, #72231978, #72231980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390539);