Finite Group Databasebeta

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Cayley diagram

C2×#21155 #6371864

order512
nilpotency class3
isoclinismisoclinic class of #3057
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21155

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2954 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)(7,8,10,9)(13,19)(14,21)(16,22)(18,24)(20,25)(23,26);(4,6)(7,9,10,8)(11,12)(13,16)(14,18)(15,17)(19,22)(20,23)(21,24)(25,26);(11,13,14,19,15,20,21,25)(12,16,18,22,17,23,24,26);(12,17)(16,23)(18,24)(22,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264586, #72264592, C2×#6381945, …
maximal normal subgroup of#72264586, #72264592, C2×#6381945, …
maximal quotient ofC2×#203556, #72320374, #72321638

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279060);