C₂×#2805666 #72331809

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

Aliases

direct product

C₂×#2805666

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

46505 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)(20,24)(21,26);(3,4)(5,6)(7,8)(9,10)(11,12,14,16)(13,15,18,17)(19,20)(21,25)(22,26)(23,24);(11,13,14,18)(12,15,16,17)(19,21,23,26)(20,25,24,22);(1,2);(3,4,5,6)(11,13)(12,15)(14,18)(16,17)(19,21)(20,22)(23,26)(24,25);(3,4)(5,6)(11,12)(13,17)(14,16)(15,18)(19,22)(20,26)(21,24)(23,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297936, C₂×#72323308, C₂×#72323306
maximal normal subgroup of	C₂×#72297936, C₂×#72323308, C₂×#72323306

GAP

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

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

C2×#2805666 #72331809