C₂×#126851 #72321364

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#126851

Subgroups

	subgroup	G/H
center	C₂×C₄	#1736
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	#7895
prime radical	C₂²×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(4,10,13,14,15,27,16,22)(6,11)(7,24)(12,28,17,32,30,26,31,29)(18,23)(19,33);(1,2);(3,4,11,14,8,13,24,27,9,15,6,22,25,16,7,10)(5,17,18,28,20,30,33,32,21,31,23,26,34,12,19,29);(3,5)(4,12,15,30)(6,23)(7,19)(8,20)(9,21)(10,26,27,28)(11,18)(13,17,16,31)(14,29,22,32)(24,33)(25,34)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224951, C₂×#72224963, C₂×#72224957
maximal normal subgroup of	C₂×#72224951, C₂×#72224963, C₂×#72224957

GAP

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

gap> G := Group((4,10,13,14,15,27,16,22)(6,11)(7,24)(12,28,17,32,30,26,31,29)(18,23)(19,33), (1,2), (3,4,11,14,8,13,24,27,9,15,6,22,25,16,7,10)(5,17,18,28,20,30,33,32,21,31,23,26,34,12,19,29), (3,5)(4,12,15,30)(6,23)(7,19)(8,20)(9,21)(10,26,27,28)(11,18)(13,17,16,31)(14,29,22,32)(24,33)(25,34));

C2×#126851 #72321364