Finite Group Databasebeta

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Cayley diagram

C2×#12795 #493695

order512
nilpotency class4
isoclinismisoclinic class of #11502
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#12795

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240316, #72240317, C2×#512030, …
maximal normal subgroup of#72240316, #72240317, C2×#512030, …
maximal quotient of#72261315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400891);