C₂×#989161 #72325109

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#989161

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

47851 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)(13,15)(14,16)(17,18)(19,20,23,26)(21,22,25,24);(3,4,6,5)(12,13)(16,17)(19,21)(20,24)(22,26)(23,25);(7,8)(9,10)(21,25)(22,24);(4,5)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,21)(20,24)(22,26)(23,25);(1,2);(3,5,6,4)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22,23,24)(20,25,26,21)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232165, C₂×#72232134
maximal normal subgroup of	C₂×#72232077, C₂×#72232165, 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)(13,15)(14,16)(17,18)(19,20,23,26)(21,22,25,24), (3,4,6,5)(12,13)(16,17)(19,21)(20,24)(22,26)(23,25), (7,8)(9,10)(21,25)(22,24), (4,5)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,21)(20,24)(22,26)(23,25), (1,2), (3,5,6,4)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22,23,24)(20,25,26,21));

C2×#989161 #72325109