C₂×#520465 #72340060

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#520465

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

249501 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)(11,12)(13,14)(15,16)(17,22)(18,21)(19,20);(15,17,20,18)(16,21,19,22);(7,8)(9,10)(15,18,20,17)(16,21,19,22);(4,6)(8,10)(11,12,13,14)(15,19,20,16)(17,21,18,22);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337324
maximal normal subgroup of	C₂×#72337324
maximal quotient of	#72340068, #72340071, C₂×#72337385

GAP

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

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

C2×#520465 #72340060