C₂×#836622 #72332543

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

Aliases

direct product

C₂×#836622

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

39253 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)(21,26)(22,25)(23,24);(4,6)(7,8)(9,10)(11,12,15,13)(14,16,18,17)(20,24)(25,26);(4,6)(19,21,23,22)(20,25,24,26);(1,2);(3,4,5,6)(11,13,15,12)(14,16,18,17)(19,22,23,21)(20,25,24,26);(11,14)(12,16)(13,17)(15,18)(19,20)(21,26)(22,25)(23,24)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323422, C₂×#72323391, C₂×#72323392
maximal normal subgroup of	C₂×#72323422, C₂×#72323391, C₂×#72323392

GAP

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

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

C2×#836622 #72332543