C₂×#650397 #72324410

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#650397

Subgroups

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

52705 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232044, C₂×#72232062, C₂×#72232036
maximal normal subgroup of	C₂×#72232044, C₂×#72232062, C₂×#72232036

GAP

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

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

C2×#650397 #72324410