#120084

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 472 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(2,3)(5,6)(7,14)(8,12)(9,11)(10,15)(13,17)(16,20)(18,19)(21,22,26,30,27,32,25,31)(23,28,35,43,36,44,34,48)(24,37,39,29,40,46,41,47)(33,45,49,42,50,52,38,51);(1,2,4,3)(5,7,8,15)(6,10,12,14)(9,18,17,16)(11,20,13,19)(21,23)(22,28)(24,38)(25,34)(26,35)(27,36)(29,45)(30,43)(31,48)(32,44)(33,39)(37,51)(40,49)(41,50)(42,46)(47,52);(1,3,4,2)(6,11)(9,17)(10,19)(12,13)(14,20)(16,18)(21,24,27,40)(22,29,32,47)(23,33,36,50)(25,41,26,39)(28,42,44,51)(30,46,31,37)(34,38,35,49)(43,52,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,e^c=eh,f^c=fhi

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, 27280);