C₂×#4928780 #72320392

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄³

Aliases

direct product

C₂×#4928780

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

78385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72320249, C₂²×#4928780
maximal normal subgroup of	C₂×#72320249, C₂²×#4928780
maximal quotient of	C₂×#72278310, C₂²×#4928780

GAP

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

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

C2×#4928780 #72320392