C₂×#651706 #72325477

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

Aliases

direct product

C₂×#651706

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

42197 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232184, C₂×#72232157, C₂×#72232165
maximal normal subgroup of	C₂×#72232184, C₂×#72232157, C₂×#72232165

GAP

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

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

C2×#651706 #72325477