C₂×#481421 #72285956

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#481421

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×D₈×#41	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216771, C₂×#72216810, C₂×#72216916, …
maximal normal subgroup of	C₂×#72216771, C₂×#72216810, C₂×#72216916, …
maximal quotient of	C₂×D₄×#1333, C₂×#72225044, #72319487, …

GAP

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

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

C2×#481421 #72285956