C₂×#1418479 #72296435

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1418479

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

51351 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232174, C₂×#72282328, C₂×#72298088, …
maximal normal subgroup of	C₂×#72232174, C₂×#72282328, C₂×#72298088, …
maximal quotient of	C₂×#72278117

GAP

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

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

C2×#1418479 #72296435