Finite Group Databasebeta

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Cayley diagram

C2×#20662 #6370811

order512
nilpotency class3
isoclinismisoclinic class of #3057
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#20662

Subgroups

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

4930 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)(5,6)(7,8,9,10)(12,16)(14,21)(17,22)(18,23)(20,25)(24,26);(11,12,14,16,15,18,21,23)(13,17,20,22,19,24,25,26);(4,6)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26);(13,19)(17,24)(20,25)(22,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72307045, C2×#6372955, #72307051, …
maximal normal subgroup of#72307045, C2×#6372955, #72307051, …
maximal quotient ofC2×#189509, #72320257, #72321563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278007);