C₂×#140751 #72286533

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂×#140751

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂³×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72227144, C₂×#72227150, C₂×#72227128, …
maximal normal subgroup of	C₂×#72227144, C₂×#72227150, C₂×#72227128, …

GAP

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

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

C2×#140751 #72286533