#121747

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#390
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14,13,16,11,15)(18,23,26,25)(19,27,30,29);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(15,16)(17,18)(19,28)(20,25)(21,23)(22,26)(24,29)(27,32)(30,31);(1,4,5,3)(2,6,7,8)(10,14)(11,12)(15,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,28)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28943);