C₂×#648371 #72299281

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

Aliases

direct product

C₂×#648371

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

65701 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213999, C₂×#72297195, C₂×#72298085, …
maximal normal subgroup of	C₂×#72213999, C₂×#72297195, C₂×#72298085, …

GAP

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

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

C2×#648371 #72299281