C₂⁴×#476 #72305083

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂⁴×#476

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

113733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

(9,10,13,15)(11,14,12,16);(1,2)(3,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,21)(19,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,14,15,16)(17,18,20,22)(19,24,23,21);(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	#72305093, C₂⁴×#3109, #72314439, …
maximal normal subgroup of	#72305093, C₂⁴×#3109, #72314439, …

GAP

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

gap> G := Group((9,10,13,15)(11,14,12,16), (1,2)(3,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,21)(19,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,14,15,16)(17,18,20,22)(19,24,23,21), (1,2), (3,4), (5,6), (7,8));

C24×#476 #72305083