C₂×#1420335 #72325331

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#1420335

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

43085 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)(11,12,15,13)(14,16,18,17)(19,20)(21,25)(22,23)(24,26);(3,4,6,5)(8,10)(12,13)(16,17)(19,21)(20,24)(22,25)(23,26);(4,5)(8,10)(14,18)(16,17);(4,5)(7,8)(9,10)(11,13,15,12)(14,16,18,17)(19,22,23,20)(21,25,26,24);(1,2);(3,5,6,4)(11,14)(12,16)(13,17)(15,18)(19,21)(20,25)(22,24)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232078, C₂×#72232089, C₂×#72232187, …
maximal normal subgroup of	C₂×#72232078, C₂×#72232089, C₂×#72232187, …

GAP

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

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

C2×#1420335 #72325331