#121007

order	512
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fi,g^b=gi,f^c=fhi,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, 28203);