#120083

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

516 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,e^c=eh,f^c=fhi

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, 27279);