C₂×#651107 #72324046

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

Aliases

direct product

C₂×#651107

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48657 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)(19,20)(21,24)(22,25)(23,26);(7,8)(9,10)(11,12,15,14)(13,16,17,18);(4,6)(19,21)(20,22)(23,25)(24,26);(1,2);(4,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,12)(13,16,17,18)(19,22,23,24)(20,25,26,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323251, C₂×#72300569, C₂×#72323274, …
maximal normal subgroup of	C₂×#72323251, C₂×#72300569, C₂×#72323274, …

GAP

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

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

C2×#651107 #72324046