#111289

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

760 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 18485);