C₂×#648366 #72298697

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#648366

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

79045 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233551, C₂×#72233568, C₂×#72233567, …
maximal normal subgroup of	C₂×#72233551, C₂×#72233568, C₂×#72233567, …
maximal quotient of	C₂×#72298477, C₂²×#648366

GAP

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

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

C2×#648366 #72298697