Finite Group Databasebeta

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Cayley diagram

C2×#24260 #6442719

order512
nilpotency class3
isoclinismisoclinic class of #23900
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C211

Aliases

direct productC2×#24260

Subgroups

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

6714 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)(10,13)(12,14);(11,14);(3,5,6,4)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6454041, C2×#248455, C2×D4×B(4,𝔽2), …
maximal normal subgroup ofC2×#6454041, C2×#248455, C2×D4×B(4,𝔽2), …
maximal quotient of#72236890, #72236900, #72249446, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349915);