C₂×#7631575 #72304178

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7631575

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

97581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72257036
maximal normal subgroup of	C₂×#72257036

GAP

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

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

C2×#7631575 #72304178