C₂×#122468 #72322355

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#122468

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂³	C₂×D₄⋊C₈
prime radical	C₂×#3169	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1529 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72286692, C₂×#72286314, C₂×#72286672
maximal normal subgroup of	C₂×#72286692, C₂×#72286314, C₂×#72286672
maximal quotient of	#72322430, #72322487

GAP

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

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

C2×#122468 #72322355