C₂⁴×#459 #72305097

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁴×#459

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁷
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

169285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72305100, C₂⁴×#3109, #72314446, …
maximal normal subgroup of	#72305100, C₂⁴×#3109, #72314446, …

GAP

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

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

C24×#459 #72305097