#119904

order	512
nilpotency class	3
isoclinism	isoclinic class of #10549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

579 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:

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

Polycyclic presentation:

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

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, 27100);