C₂×#1083405 #72325611

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1083405

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

39783 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232189, C₂×#72232183, C₂×#72232169, …
maximal normal subgroup of	C₂×#72232189, C₂×#72232183, C₂×#72232169, …

GAP

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

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

C2×#1083405 #72325611