C₂×#797415 #72325098

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

Aliases

direct product

C₂×#797415

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

41367 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232190, C₂×#72232160, C₂×#72232164
maximal normal subgroup of	C₂×#72232190, C₂×#72232160, C₂×#72232164

GAP

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

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

C2×#797415 #72325098