C₂×#7654748 #72298409

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7654748

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

89793 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)(8,11)(9,13)(17,21)(20,22);(3,4)(5,6)(8,11)(12,14)(16,19)(17,21);(7,8,10,11)(9,12,13,14)(15,16)(17,20)(18,19)(21,22);(7,8,10,11)(9,12,13,14)(15,17,18,21)(16,20,19,22);(7,8)(9,14)(10,11)(12,13);(7,9)(8,12)(10,13)(11,14);(1,2)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234453
maximal normal subgroup of	C₂×#72234453
maximal quotient of	C₂×#72297599, C₂×#72298096

GAP

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

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

C2×#7654748 #72298409