Finite Group Databasebeta

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Cayley diagram

C2×#25771 #6443345

order512
nilpotency class3
isoclinismisoclinic class of #24404
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210×C4

Aliases

direct productC2×#25771

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

7308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 14 points:

(1,2);(4,5)(7,8)(9,11)(10,12)(13,14);(3,4,6,5)(7,9)(8,11)(10,14)(12,13);(7,8)(9,13,14,11)(10,12);(3,5,6,4)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×B(4,𝔽2), #72240665, C2×#6533135, …
maximal normal subgroup ofC2×D4×B(4,𝔽2), #72240665, C2×#6533135, …
maximal quotient of#72237546, #72237552, #72249448, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350541);