#109824

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,26,23,27,21,36)(19,28,31,40,32,39,30,24)(20,33,34,25,35,41,37,42)(29,44,45,48,47,38,46,43);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19)(18,24,27,40)(20,29)(21,30)(22,31)(23,32)(25,38,42,44)(26,28,36,39)(33,48,41,43)(34,45)(35,47)(37,46);(4,14)(7,12)(9,16)(10,13)(17,20,21,37,23,35,22,34)(18,25,36,33,27,42,26,41)(19,29,30,46,32,47,31,45)(24,38,39,48,40,44,28,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ehi

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, 17020);