C₂×#567026 #72297360

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

Aliases

direct product

C₂×#567026

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

40113 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)(7,8,9,10)(11,12,14,15)(13,16,18,17);(3,4)(5,6)(19,20)(21,24)(22,25)(23,26);(4,6)(12,15)(16,17)(19,21)(20,23)(22,26)(24,25);(3,4)(5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,25)(24,26);(1,2)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232107, C₂×#72232169, C₂×#72232121
maximal normal subgroup of	C₂×#72232107, C₂×#72232169, C₂×#72232121
maximal quotient of	C₂×#72278130

GAP

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

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

C2×#567026 #72297360