C₂×#836596 #72324050

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

Aliases

direct product

C₂×#836596

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

50231 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)(13,15)(14,16)(17,18);(7,8)(9,10)(19,20,23,22)(21,24,25,26);(4,6)(12,13)(14,18);(1,2);(4,6)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,18)(12,16,13,17)(19,22,23,20)(21,24,25,26)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323248, C₂×#72323247, C₂×#72323274
maximal normal subgroup of	C₂×#72323248, C₂×#72323247, C₂×#72323274

GAP

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

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

C2×#836596 #72324050