Finite Group Databasebeta

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Cayley diagram

C2×#23356 #6407624

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#23356

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC22×#127C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(3,4)(5,9)(6,10)(7,8)(11,12)(13,17)(14,16)(15,18)(19,20,23,24,22,25,21,26);(4,8)(5,6)(13,18)(15,17);(3,5,7,6)(4,9,8,10)(11,13,14,18)(12,15,16,17)(19,21,22,23)(20,24,25,26);(3,6,7,5)(4,9,8,10)(12,16)(15,17)(20,25)(24,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6407637, #72264750, #72264760
maximal normal subgroup ofC2×#6407637, #72264750, #72264760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314820);