C₂×#1495035 #72332534

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1495035

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47709 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323422, C₂×#72323415, C₂×#72323416
maximal normal subgroup of	C₂×#72323422, C₂×#72323415, C₂×#72323416

GAP

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

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

C2×#1495035 #72332534