C₂×#2120602 #72296839

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#2120602

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41731 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278635
maximal normal subgroup of	C₂×#72278635
maximal quotient of	C₂×#72278761, C₂×#72278726

GAP

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

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

C2×#2120602 #72296839