#243954

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

Aliases

No aliases for this group.

Subgroups

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

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

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, 151150);