C₂×#512038 #72307126

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#512038

Subgroups

	subgroup	G/H
center	C₂×C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁵
socle	C₂²	#32976
prime radical	C₂×#3481	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

5216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307130, #72307167, C₂×#72231599, …
maximal normal subgroup of	#72307130, #72307167, C₂×#72231599, …

GAP

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

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

C2×#512038 #72307126