C₂×#1088915 #72324392

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#1088915

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

57547 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72219264, C₂×#72232044, C₂×#72232052
maximal normal subgroup of	C₂×#72219264, C₂×#72232044, C₂×#72232052

GAP

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

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

C2×#1088915 #72324392