#141890

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=ei,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49086);