C₂×#17085 #844845

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#17085

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4482 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 4482 conjugacy classes of subgroups — too many to draw. Only its 986 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#17085⋊C₃
maximal subgroup of	#72267035, #72267036, C₂×#17085⋊C₃
maximal normal subgroup of	#72267035, #72267036, C₂×#17085⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752041);