Finite Group Databasebeta

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Cayley diagram

C2×#23551 #6431974

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#23551

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutator, Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(7,8,11,13)(9,14,10,12)(15,16)(17,24)(18,22)(19,27)(20,23)(21,25)(26,29)(28,30);(3,4)(5,6)(7,9,11,10)(8,12,13,14);(4,6)(7,10,11,9)(8,12,13,14)(15,17)(16,21)(18,25)(19,28)(20,26)(22,29)(23,30)(24,27);(15,18,20,19)(16,22,23,27)(17,25,26,28)(21,29,30,24)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6453019, C2×#6445496, C2×#6432017
maximal normal subgroup ofC2×#6453019, C2×#6445496, C2×#6432017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339170);