C₂×#519568 #72302454

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#519568

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₄×D₄×#120, C₂×#72280257, C₂×#72280004, …
maximal normal subgroup of	C₂×C₄×D₄×#120, C₂×#72280257, C₂×#72280004, …
maximal quotient of	C₂²×#519568, C₂×#72298440, C₂×#72317640

GAP

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

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

C2×#519568 #72302454