#118509

order	512
nilpotency class	3
isoclinism	isoclinic class of #118505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25705);