Finite Group Databasebeta

← Prev (#7626192) Next (#7626194) →

Cayley diagram

C4×#3544 #7626193

order512
nilpotency class2
isoclinismisoclinic class of #3532
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC4×#3544

Subgroups

subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3544C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

6994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4);(6,9)(7,11)(14,18)(16,24)(19,21)(22,28);(6,9)(10,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(5,6,8,9)(7,10,11,12);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#62759, #72315206, #72315209
maximal normal subgroup ofC4×#62759, #72315206, #72315209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533389);