C₂⁴×#432 #72305123

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂⁴×#432

Subgroups

	subgroup	G/H
center	C₂⁵×C₄	C₂³
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂×C₄	C₂⁷
socle	C₂⁶	C₂²×C₄
prime radical	C₂⁷	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

114805 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)(11,14)(12,13)(15,16)(18,21)(22,24);(10,13)(14,16)(17,18,20,22,19,21,23,24);(1,2);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(18,21)(22,24);(3,4);(5,6);(7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#3109, #72305155, #72305160, …
maximal normal subgroup of	C₂⁴×#3109, #72305155, #72305160, …

GAP

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

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

C24×#432 #72305123