#114931

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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

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

normal subgroups only

This group has 850 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)(5,6,9,12,10,13,8,14)(7,15,17,20,18,11,16,19)(21,22)(23,33)(24,38,42,47)(25,31)(26,30)(27,32)(28,35)(29,40,48,41)(34,43,50,52)(36,45)(37,44)(39,51,49,46);(2,4)(5,7,10,18)(6,11,13,15)(8,17,9,16)(12,20,14,19)(21,23,27,37)(22,28,32,45)(24,39,42,49)(25,36,26,35)(29,46,48,51)(30,44,31,33)(34,40,50,41)(38,43,47,52);(6,12)(8,9)(11,19)(13,14)(15,20)(16,17)(21,24)(22,29)(23,34)(25,41)(26,40)(27,42)(28,43)(30,38)(31,47)(32,48)(33,46)(35,39)(36,49)(37,50)(44,51)(45,52)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,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, 22127);