C₂×#648359 #72341796

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

Aliases

direct product

C₂×#648359

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

58749 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72260984, C₂×#72281143, C₂×#72268529
maximal normal subgroup of	C₂×#72260984, C₂×#72281143, C₂×#72268529

GAP

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

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

C2×#648359 #72341796