C₂×#7674650 #72324846

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

Aliases

direct product

C₂×#7674650

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 6206 conjugacy classes of subgroups; only the 32357 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232067
maximal normal subgroup of	C₂×#72232067
maximal quotient of	C₂×#72232144

GAP

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

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

C2×#7674650 #72324846