C₂×#647718 #72331099

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#647718

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

70669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72264092, C₂×#72323287, C₂×#72261595, …
maximal normal subgroup of	C₂×#72264092, C₂×#72323287, C₂×#72261595, …

GAP

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

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

C2×#647718 #72331099