C₂×#784923 #72325114

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

Aliases

direct product

C₂×#784923

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

66079 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)(11,12,14,13)(15,16)(17,18)(19,21)(20,22);(3,4)(5,6)(7,8,10,9)(11,12,14,13)(15,17)(16,20)(18,21)(19,22);(8,9)(12,13);(4,5)(8,9)(11,12,14,13)(15,18,19,20)(16,17,21,22);(1,2);(3,5,6,4)(7,9,10,8)(11,13,14,12)(15,17)(16,18)(19,22)(20,21)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72219261, C₂×#72232089, C₂×#72232158, …
maximal normal subgroup of	C₂×#72219261, C₂×#72232089, C₂×#72232158, …

GAP

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

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

C2×#784923 #72325114