#113055

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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

890 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

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, 20251);