C₂×#918727 #72324014

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#918727

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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297911, C₂×#72323257, C₂×#72323270
maximal normal subgroup of	C₂×#72297911, C₂×#72323257, C₂×#72323270

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

C2×#918727 #72324014