Finite Group Databasebeta

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Cayley diagram

C2×#20649 #6370790

order512
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#20649

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5194 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);(4,6)(7,8,9,10)(11,12)(13,21)(14,20)(15,17)(16,19)(18,22)(23,26)(24,25);(3,4,5,6)(12,17)(18,25)(19,20)(21,26);(11,13)(12,18)(14,22)(15,23)(16,24)(17,25)(19,21)(20,26);(11,14,15,16)(12,19,17,20)(13,22,23,24)(18,21,25,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6442664, C2×#6379066, C2×#6628934
maximal normal subgroup ofC2×#6442664, C2×#6379066, C2×#6628934

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277986);