C₂×#1310867 #72332879

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

Aliases

direct product

C₂×#1310867

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

31987 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323397, C₂×#72323408, C₂×#72323409
maximal normal subgroup of	C₂×#72323397, C₂×#72323408, C₂×#72323409

GAP

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

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

C2×#1310867 #72332879