Finite Group Databasebeta

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Cayley diagram

C2×Q8×2−1+4 #10586042

order512
nilpotency class2
isoclinismisoclinic class of D4×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C219

Aliases

direct productC2×Q8×2−1+4

Subgroups

subgroupG/H
center, socleC23C26
commutator, Frattini, cosocleC22C27
prime radicalC22×2−1+4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

33492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#11710806
maximal subgroup of#72268165, C2×#11710806, 2−1+42, …
maximal normal subgroup of#72268165, C2×#11710806, 2−1+42, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493238);