Finite Group Databasebeta

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Cayley diagram

C2×#21693 #6373729

order512
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21693

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3414 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);(4,6)(7,8,10,9)(11,12)(13,19)(14,22)(15,17)(16,21)(18,23)(20,25)(24,26);(11,13,14,21,15,20,23,26)(12,16,18,19,17,24,22,25);(3,4,5,6)(12,17)(16,24)(18,22)(19,25);(7,9,10,8)(13,20)(16,24)(19,25)(21,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6373771, #72240761, #72240768, …
maximal normal subgroup ofC2×#6373771, #72240761, #72240768, …
maximal quotient of#72308346

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280925);