C₂×#17102 #844864

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

direct product

C₂×#17102

Subgroups

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

10966 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 10966 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) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#17102⋊C₅, #72238094
maximal subgroup of	#72238070, C₂×#17102⋊C₅, C₂×#4122222, …
maximal normal subgroup of	#72238070, C₂×#17102⋊C₅, C₂×#4122222, …
maximal quotient of	C₂×#158995, C₂²×#17102, C₂×#158818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752060);