C₂×#513714 #72298744

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#513714

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁴×C₄²
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80617 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)(7,8,9,10)(16,19)(20,22);(3,4)(5,6)(7,8)(9,10)(15,16)(17,20)(18,19)(21,22);(3,4,5,6)(8,10)(11,12,14,13)(16,19)(17,21);(4,6)(7,8,9,10)(11,13,14,12)(15,17,18,21)(16,20,19,22);(3,4)(5,6)(7,8)(9,10)(15,16)(17,22)(18,19)(20,21);(1,2)

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72233567, C₂×#72233563, C₂×#72233576, …
maximal normal subgroup of	C₂×#72233567, C₂×#72233563, C₂×#72233576, …
maximal quotient of	C₂×D₄×C₂²⋊#95, C₂×#72335149, C₂²×#513714

GAP

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

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

C2×#513714 #72298744