C₂×#517125 #72334541

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

Aliases

direct product

C₂×#517125

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2613	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

73433 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72330145, C₂×C₄×#34980
maximal normal subgroup of	C₂×#72330145, C₂×C₄×#34980
maximal quotient of	C₂×#72317640

GAP

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

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

C2×#517125 #72334541