C₂×#663622 #72324447

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

Aliases

direct product

C₂×#663622

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

65901 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)(8,10)(12,15)(16,17)(19,20,22,25)(21,24,26,23);(8,10)(19,21)(20,23)(22,26)(24,25);(21,26)(23,24);(11,12)(13,17)(14,15)(16,18)(19,20)(21,24)(22,25)(23,26);(1,2);(3,4,5,6)(7,8)(9,10)(11,13,14,18)(12,16,15,17)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232040, C₂×#72232037, C₂×#72232050, …
maximal normal subgroup of	C₂×#72232040, C₂×#72232037, C₂×#72232050, …

GAP

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

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

C2×#663622 #72324447