#112842

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 816 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=h,f^2=h,g^2=hi,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, 20038);