C₂⁴×#543 #72301868

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂⁴×#543

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

622357 conjugacy classes of subgroups; only the 446037 normal subgroups are shown.

normal subgroups only

This group has 622357 conjugacy classes of subgroups — too many to draw. Only its 446037 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#3490
maximal normal subgroup of	C₂⁴×#3490

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