C₂×#1418423 #72325273

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

Aliases

direct product

C₂×#1418423

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

41793 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)(11,12,14,16)(13,15,17,18)(20,23)(24,25);(3,4)(5,6)(19,20)(21,24)(22,23)(25,26);(4,5)(8,10)(13,17)(15,18);(4,5)(7,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26);(1,2);(3,5,6,4)(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232093, C₂×#72232084, C₂×#72232175, …
maximal normal subgroup of	C₂×#72232093, C₂×#72232084, C₂×#72232175, …

GAP

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

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

C2×#1418423 #72325273