C₂×#719100 #72331093

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

Aliases

direct product

C₂×#719100

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

68113 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)(12,14)(15,16,19,21)(17,22,20,18);(8,10)(11,12)(13,14)(15,17,19,20)(16,18,21,22);(4,6)(8,10)(15,16,19,21)(17,22,20,18);(1,2);(3,4,5,6)(7,8,9,10)(15,16)(17,18)(19,21)(20,22);(15,18)(16,20)(17,21)(19,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323284, C₂×#72323281, C₂×#72261284
maximal normal subgroup of	C₂×#72323284, C₂×#72323281, C₂×#72261284

GAP

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

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

C2×#719100 #72331093