#114932

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 780 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)(5,6,10,16,11,14,9,15)(7,17,19,20,8,12,18,13)(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)(5,7,11,8)(6,12,14,17)(9,19,10,18)(13,16,20,15)(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);(5,8,11,7)(6,13,14,20)(9,18,10,19)(12,15,17,16)(21,24,27,42)(22,29,32,48)(23,34,37,50)(25,41,26,40)(28,43,45,52)(30,38,31,47)(33,46,44,51)(35,39,36,49)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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, 22128);