Finite Group Databasebeta

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Cayley diagram

C2×#31247 #7503884

order512
nilpotency class3
isoclinismisoclinic class of #30412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#31247

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265333, C2×#7504005, #72265460, …
maximal normal subgroup of#72265333, C2×#7504005, #72265460, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411080);