C₂×#1426723 #72332658

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

Aliases

direct product

C₂×#1426723

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

53171 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323417, C₂×#72323356, C₂×#72323387
maximal normal subgroup of	C₂×#72323417, C₂×#72323356, C₂×#72323387

GAP

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

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

C2×#1426723 #72332658