#243151

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 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)(3,8)(4,6)(5,7)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,15)(14,16);(1,4,5,3)(2,6,7,8)(10,13)(11,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150347);