Finite Group Databasebeta

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Cayley diagram

C4×#2433 #514831

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210×C4

Aliases

direct productC4×#2433

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2433C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

9086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

(1,2,3,4);(6,7)(10,12);(5,6,8,7)(13,14)(15,16);(9,10,11,12)(14,16);(5,7,8,6)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226824, D4×#2433, #72233680, …
maximal normal subgroup of#72226824, D4×#2433, #72233680, …
maximal quotient of#72224689, #72224884, AGL(1,𝔽5)×#2433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422027);