C₂×#850557 #72337854

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#850557

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

138105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232282, C₂×#72232336, C₂×#72232311
maximal normal subgroup of	C₂×#72232282, C₂×#72232336, C₂×#72232311

GAP

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

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

C2×#850557 #72337854