C₂×#647794 #72311233

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#647794

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

61781 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);(8,10)(16,19)(17,21);(11,12,13,14)(15,16)(17,20)(18,19)(21,22);(11,12)(13,14)(15,17,18,21)(16,20,19,22);(1,2);(3,4,5,6)(7,8,9,10)(15,16)(17,20)(18,19)(21,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249699, C₂×#72268529, C₂×#120×#123
maximal normal subgroup of	C₂×#72249699, C₂×#72268529, C₂×#120×#123

GAP

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

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

C2×#647794 #72311233