C₂×#618353 #72332520

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

Aliases

direct product

C₂×#618353

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

39507 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323385, C₂×#72323396, C₂×#72323413
maximal normal subgroup of	C₂×#72323385, C₂×#72323396, C₂×#72323413

GAP

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

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

C2×#618353 #72332520