C₂×#3874586 #72325383

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#3874586

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

52501 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232101, C₂×#72232146, C₂×#72232126
maximal normal subgroup of	C₂×#72232101, C₂×#72232146, C₂×#72232126

GAP

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

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

C2×#3874586 #72325383