C₂×#533176 #72299207

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#533176

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

59917 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298088, C₂×#72298090, C₂×#72298102, …
maximal normal subgroup of	C₂×#72298088, C₂×#72298090, C₂×#72298102, …

GAP

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

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

C2×#533176 #72299207