C₂×#1418753 #72299405

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

Aliases

direct product

C₂×#1418753

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

40201 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72281456, C₂×#72298076
maximal normal subgroup of	C₂×#72281456, C₂×#72298076

GAP

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

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

C2×#1418753 #72299405