#141053

order	512
nilpotency class	4
isoclinism	isoclinic class of #141021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,22,26,21,38,36,27,23,29,37,45,24,28,20,35)(19,30,31,25,40,42,32,41,34,47,48,43,33,46,39,44);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,24,40,23,34,21,33)(18,25,38,44,29,43,28,41)(20,31,37,32,36,48,22,39)(26,30,27,46,45,47,35,42);(2,7)(6,14)(8,12)(9,15)(17,20,24,37,23,36,21,22)(18,26,38,27,29,45,28,35)(19,31,40,32,34,48,33,39)(25,30,44,46,43,47,41,42)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=eg,d^2=ghi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48249);