Finite Group Databasebeta

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Cayley diagram

C2×#17671 #6343793

order512
nilpotency class3
isoclinismisoclinic class of #17577
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#17671

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC22×#532C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

4288 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);(3,4,5,8,6,7,9,10)(11,12)(13,20,22,25)(14,23)(15,18)(16,19)(17,21,26,24);(4,7)(5,9)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20);(4,7)(8,10)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238926, #72238927, C2×#186250, …
maximal normal subgroup of#72238926, #72238927, C2×#186250, …
maximal quotient of#72238954

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250989);