C₂×#918774 #72299250

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#918774

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

28509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297354, C₂×#72297416, C₂×#72297599
maximal normal subgroup of	C₂×#72297354, C₂×#72297416, C₂×#72297599

GAP

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

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

C2×#918774 #72299250