C₂×#647660 #72310697

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#647660

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

87809 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)(9,10)(12,13)(15,16)(17,21)(18,19)(20,22);(4,6)(8,10);(4,6)(8,10)(16,19)(17,22);(1,2)(7,8)(9,10)(11,12,14,13)(15,17)(16,20)(18,22)(19,21);(7,8)(9,10)(11,13,14,12)(15,17)(16,20)(18,22)(19,21);(3,4)(5,6)(7,8)(9,10)(15,16)(17,20)(18,19)(21,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72240172, C₂×#72240157, C₂×#72264092, …
maximal normal subgroup of	C₂×#72240172, C₂×#72240157, C₂×#72264092, …
maximal quotient of	C₂×#72310627, C₂×#72310661

GAP

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

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

C2×#647660 #72310697