C₂×#523559 #72323889

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#523559

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49477 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323252, C₂×#72323267, C₂×#72323253
maximal normal subgroup of	C₂×#72323252, C₂×#72323267, C₂×#72323253

GAP

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

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

C2×#523559 #72323889