C₂×#622391 #72331820

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#622391

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2613	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

46635 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323299, C₂×#72323302
maximal normal subgroup of	C₂×#72323299, C₂×#72323302

GAP

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

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

C2×#622391 #72331820