C₂×#850576 #72324026

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#850576

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

41173 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297911, C₂×#72323271, C₂×#72297746
maximal normal subgroup of	C₂×#72297911, C₂×#72323271, C₂×#72297746

GAP

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

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

C2×#850576 #72324026