Finite Group Databasebeta

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Cayley diagram

C4×#3459 #7625314

order512
nilpotency class2
isoclinismisoclinic class of #534
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC4×#3459

Subgroups

subgroupG/H
centerC2×C42C24
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3459C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C23 ▷ C1

8342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2,3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20);(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(14,17)(18,20);(6,10)(9,11);(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282043, #72282116, #72282131
maximal normal subgroup of#72282043, #72282116, #72282131

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532510);