C₂×#7684911 #72279804

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

Aliases

direct product

C₂×#7684911

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

At least 9754 conjugacy classes of subgroups; only the 32125 normal subgroups are shown.

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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72262511
maximal normal subgroup of	C₂×#72262511

GAP

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

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

C2×#7684911 #72279804