C₂×#740202 #72325271

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#740202

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(8,10)(11,12,15,16)(13,14,17,18)(19,20)(21,22)(23,24)(25,26);(3,4,6,5)(8,10)(12,16)(13,17)(20,24)(21,25);(12,16)(14,18);(4,5)(7,8,9,10)(11,13,15,17)(12,14,16,18)(19,21,23,25)(20,22,24,26);(1,2);(3,5,6,4)(11,14)(12,17)(13,16)(15,18)(19,22)(20,21)(23,26)(24,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232084, C₂×#72232127, C₂×#72232160
maximal normal subgroup of	C₂×#72232084, C₂×#72232127, C₂×#72232160

GAP

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

gap> G := Group((8,10)(11,12,15,16)(13,14,17,18)(19,20)(21,22)(23,24)(25,26), (3,4,6,5)(8,10)(12,16)(13,17)(20,24)(21,25), (12,16)(14,18), (4,5)(7,8,9,10)(11,13,15,17)(12,14,16,18)(19,21,23,25)(20,22,24,26), (1,2), (3,5,6,4)(11,14)(12,17)(13,16)(15,18)(19,22)(20,21)(23,26)(24,25));

C2×#740202 #72325271