#138555

order	512
nilpotency class	4
isoclinism	isoclinic class of #138552
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19225	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45751);