C₂×#652612 #72299286

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#652612

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297416, C₂×#72297560, C₂×#72261412
maximal normal subgroup of	C₂×#72297416, C₂×#72297560, C₂×#72261412

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,6)(16,18), (7,8)(9,10)(11,12,13,14)(15,16)(17,18), (4,6)(7,8,9,10)(11,12,13,14), (4,6)(11,12,13,14), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14), (1,2));

C2×#652612 #72299286