C₂⁴×#381 #72291299

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂⁴×#381

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂³
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

522133 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:

(18,21)(22,24);(10,12)(13,14,15,16)(17,18)(19,22)(20,21)(23,24);(3,4);(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,22,21,24);(5,6);(7,8);(1,2)(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×#2339, C₂⁴×#2404, C₂⁴×#2613, …
maximal normal subgroup of	C₂⁴×#2339, C₂⁴×#2404, C₂⁴×#2613, …

GAP

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

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

C24×#381 #72291299