C₂×#7708685 #72299012

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

Aliases

direct product

C₂×#7708685

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

85889 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297217
maximal normal subgroup of	C₂×#72297217
maximal quotient of	C₂×#72297593

GAP

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

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

C2×#7708685 #72299012