C₂×#94425 #72250451

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#94425

Subgroups

	subgroup	G/H
center, socle	C₂²	#7304
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂×#535	C₂³
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250459, #72250468, #72250474, …
maximal normal subgroup of	#72250459, #72250468, #72250474, …

GAP

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

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

C2×#94425 #72250451