C₂×#687728 #72299459

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#687728

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

49401 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)(8,10)(12,14)(16,19)(20,22);(4,6)(11,12)(13,14)(17,21)(20,22);(4,6)(7,8,9,10);(3,4)(5,6)(15,16,18,19)(17,20,21,22);(3,4)(5,6)(15,17)(16,20)(18,21)(19,22);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298093, C₂×#72298084, C₂×#72261010
maximal normal subgroup of	C₂×#72298093, C₂×#72298084, C₂×#72261010

GAP

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

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

C2×#687728 #72299459