C₂×#4035576 #72337866

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

Aliases

direct product

C₂×#4035576

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

44457 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232283, C₂×#72232334, C₂×#72232305
maximal normal subgroup of	C₂×#72232283, C₂×#72232334, C₂×#72232305

GAP

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

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

C2×#4035576 #72337866