C₂×#991218 #72325232

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#991218

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

47295 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)(7,8)(9,10)(11,12,16,15)(13,18,14,17)(19,20,23,26)(21,24,25,22);(3,4)(5,6)(7,8)(9,10)(19,21)(20,24)(22,26)(23,25);(4,5)(11,13,16,14)(12,17,15,18)(19,22,23,24)(20,25,26,21);(4,5)(8,10)(11,14,16,13)(12,18,15,17)(19,22)(20,21)(23,24)(25,26);(1,2);(3,5,6,4)(7,8)(9,10)(11,15,16,12)(13,18,14,17)(19,20)(21,22)(23,26)(24,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232168, C₂×#72232118, C₂×#72232134
maximal normal subgroup of	C₂×#72232168, C₂×#72232118, C₂×#72232134

GAP

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

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

C2×#991218 #72325232