Finite Group Databasebeta

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Cayley diagram

C2×#12316 #488880

order512
nilpotency class4
isoclinismisoclinic class of #12316
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#12316

Subgroups

subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225208, C2×D8×D16, #72226892
maximal normal subgroup of#72225208, C2×D8×D16, #72226892
maximal quotient ofC2×#123275, #72225567, #72225571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396076);