C₂×#687608 #72299271

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#687608

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

65197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297589, C₂×#72298093, C₂×#72298095, …
maximal normal subgroup of	C₂×#72297589, C₂×#72298093, C₂×#72298095, …

GAP

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

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

C2×#687608 #72299271