C₂×#544418 #72332907

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#544418

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

39381 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)(15,17)(16,18)(19,20)(21,26)(22,24)(23,25);(11,13)(12,16)(14,17)(15,18);(1,2);(4,6)(11,12,15,17)(13,16,18,14)(19,21,23,22)(20,24,25,26);(11,12)(13,16)(14,18)(15,17)(19,20)(21,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,16)(12,13,17,18)(19,22,23,21)(20,24,25,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323391, C₂×#72323396, C₂×#72323421
maximal normal subgroup of	C₂×#72323391, C₂×#72323396, C₂×#72323421

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

C2×#544418 #72332907