C₂×#122924 #72231986

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#122924

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂³	C₂×#334
prime radical	C₂⁴	#334
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(4,8,9,14)(5,11)(6,13)(10,17,16,15)(12,18)(19,20,22,27,24,28,33,40,23,26,32,38,34,39,44,48)(21,25,30,35,31,37,43,46,29,36,41,45,42,47,49,50);(19,21)(20,25)(22,30)(23,29)(24,31)(26,36)(27,35)(28,37)(32,41)(33,43)(34,42)(38,45)(39,47)(40,46)(44,49)(48,50);(3,4,5,8,6,10,11,15,7,9,12,14,13,16,18,17)(20,26)(21,29)(27,38)(28,39)(30,41)(31,42)(40,48)(43,49)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231934, C₂×#72286643, C₂×#72286314
maximal normal subgroup of	C₂×#72231934, C₂×#72286643, C₂×#72286314
maximal quotient of	#72231987, #72231989, #72234316

GAP

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

gap> G := Group((1,2), (4,8,9,14)(5,11)(6,13)(10,17,16,15)(12,18)(19,20,22,27,24,28,33,40,23,26,32,38,34,39,44,48)(21,25,30,35,31,37,43,46,29,36,41,45,42,47,49,50), (19,21)(20,25)(22,30)(23,29)(24,31)(26,36)(27,35)(28,37)(32,41)(33,43)(34,42)(38,45)(39,47)(40,46)(44,49)(48,50), (3,4,5,8,6,10,11,15,7,9,12,14,13,16,18,17)(20,26)(21,29)(27,38)(28,39)(30,41)(31,42)(40,48)(43,49));

C2×#122924 #72231986