#115436

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

882 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,g^b=gh,d^c=di,e^c=ei,g^c=gi

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, 22632);