C₂×#479968 #72318427

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#479968

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂²×#97
prime radical	C₂²×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 50 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216898, C₂×#72216668, C₂×#72224960, …
maximal normal subgroup of	C₂×#72216898, C₂×#72216668, C₂×#72224960, …

GAP

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

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

C2×#479968 #72318427