C₂×#647925 #72323955

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#647925

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38789 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323251, C₂×#72281404
maximal normal subgroup of	C₂×#72323251, C₂×#72281404

GAP

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

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

C2×#647925 #72323955