Finite Group Databasebeta

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Cayley diagram

C2×#17501 #6343008

order512
nilpotency class3
isoclinismisoclinic class of #3186
automorphismAut(G) order 2097152
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#17501

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#186252, #72238868, #72238871
maximal normal subgroup ofC2×#186252, #72238868, #72238871
maximal quotient of#72238892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250204);