#139404

order	512
nilpotency class	4
isoclinism	isoclinic class of #139358
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,6,8,5,3,4,7)(9,10)(11,22)(12,28,31,46)(13,32)(14,20)(15,38)(16,21)(17,25)(18,30,45,37)(19,34)(23,48)(24,55,58,71)(26,50)(27,52)(29,44,33,47)(35,39)(36,51)(40,67,70,72)(41,61)(42,53)(43,62)(49,57,68,59)(54,64,65,66)(56,63,60,69);(1,3,5,2)(4,7,6,8)(9,11,23,53,15,26,13,25,16,27,34,61,14,36,35,62)(10,17,39,50,20,42,19,22,21,43,32,51,38,41,48,52)(12,24,54,67,30,59,33,60,31,58,65,72,37,57,29,56)(18,40,66,71,46,69,47,49,45,70,64,55,28,63,44,68);(4,6)(7,8)(9,12)(10,18)(11,24)(13,33)(14,37)(15,30)(16,31)(17,40)(19,47)(20,46)(21,45)(22,49)(23,54)(25,60)(26,59)(27,58)(28,38)(29,35)(32,64)(34,65)(36,57)(39,66)(41,63)(42,69)(43,70)(44,48)(50,71)(51,55)(52,68)(53,67)(56,62)(61,72)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46600);