C₂×#734802 #72324442

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#734802

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

51585 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232044, C₂×#72232041
maximal normal subgroup of	C₂×#72232044, C₂×#72232041

GAP

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

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

C2×#734802 #72324442