C₂×#151059 #72321112

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#151059

Subgroups

	subgroup	G/H
center, socle	C₂²	#12030
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	#346	C₂⁴
prime radical	C₂⁴	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×C₄×M₃₂ ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214568, C₂×#72225114, C₂×#72224979
maximal normal subgroup of	C₂×#72214568, C₂×#72225114, C₂×#72224979

GAP

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

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

C2×#151059 #72321112