#112837

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 878 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20033);