#138124

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

343 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=h,d^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45320);