C₂×#494184 #72226762

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

Aliases

direct product

C₂×#494184

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#117
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	C₂×#432
prime radical	C₂⁵	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(4,5)(7,8,9,14,10,13,17,20,11,12,16,15,18,19,22,21);(8,12)(9,16)(13,19)(17,22);(3,4,6,5)(8,12)(9,16)(13,19)(17,22);(1,2)(8,12)(13,19)(14,15)(20,21);(3,5,6,4)(8,13,12,19)(14,20,15,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72226767
maximal normal subgroup of	C₂×#72226767

GAP

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

gap> G := Group((4,5)(7,8,9,14,10,13,17,20,11,12,16,15,18,19,22,21), (8,12)(9,16)(13,19)(17,22), (3,4,6,5)(8,12)(9,16)(13,19)(17,22), (1,2)(8,12)(13,19)(14,15)(20,21), (3,5,6,4)(8,13,12,19)(14,20,15,21));

C2×#494184 #72226762