C₂⁴×#399 #72302208

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁴×#399

Subgroups

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

300501 conjugacy classes of subgroups; only the 66933 normal subgroups are shown.

normal subgroups only

This group has 300501 conjugacy classes of subgroups — too many to draw. Only its 66933 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) = 512

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#2397, C₂×#72280204, C₂×#72279950, …
maximal normal subgroup of	C₂⁴×#2397, C₂×#72280204, C₂×#72279950, …
maximal quotient of	C₂³×#14734

GAP

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

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