C₂×#519564 #72334646

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

Aliases

direct product

C₂×#519564

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

73385 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233709, C₂×#72233707, C₂×#72233715
maximal normal subgroup of	C₂×#72233709, C₂×#72233707, C₂×#72233715
maximal quotient of	C₂×#72334408

GAP

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

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

C2×#519564 #72334646