C₂×#3962538 #72323911

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

Aliases

direct product

C₂×#3962538

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

44141 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)(11,12,14,16)(13,15,17,18)(20,21)(22,26);(8,10)(13,17)(15,18)(22,26)(24,25);(7,8)(9,10)(19,20,23,21)(22,24,26,25);(3,4,5,6)(11,12)(13,18)(14,16)(15,17)(19,21,23,20)(22,25,26,24);(1,2);(3,4)(5,6)(11,13,14,17)(12,15,16,18)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323249, C₂×#72232167, C₂×#72323270, …
maximal normal subgroup of	C₂×#72323249, C₂×#72232167, C₂×#72323270, …

GAP

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

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

C2×#3962538 #72323911