C₂×#217304 #72321986

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#217304

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂²×C₄⋊C₄
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72320417
maximal normal subgroup of	C₂×#72320417
maximal quotient of	C₂×#72320519

GAP

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

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

C2×#217304 #72321986