Finite Group Databasebeta

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Cayley diagram

C4×#3498 #7625667

order512
nilpotency class2
isoclinismisoclinic class of #3484
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC4×#3498

Subgroups

subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3498C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

12474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 16 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214822, AGL(1,𝔽5)×#3498, C4×#23872, …
maximal normal subgroup of#72214822, C4×#23872, #72224762, …
maximal quotient ofAGL(1,𝔽5)×#3498, #72224778, #261×#3498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532863);