C₂×#539756 #72337944

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#539756

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

54345 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)(7,8,10,9)(11,12)(13,18)(14,16)(15,17)(19,20,23,25)(21,26,22,24);(11,13,15,16)(12,14,17,18)(19,21,23,22)(20,24,25,26);(3,4,6,5)(7,8,10,9)(19,20)(21,26)(22,24)(23,25);(3,5,6,4)(7,9,10,8)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,26)(23,25);(1,2);(11,14)(12,16)(13,17)(15,18)(19,22,23,21)(20,24,25,26)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232339, C₂×#72210496
maximal normal subgroup of	C₂×#72232339, C₂×#72210496

GAP

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

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

C2×#539756 #72337944