C₂⁴×#3490 #72301869

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴

Aliases

direct product

C₂⁴×#3490

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

11969128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#62817
maximal normal subgroup of	C₂⁴×#62817

GAP

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

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

C24×#3490 #72301869