#316749

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

629 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18,20,23)(19,25,26,32)(21,29,28,24)(22,30,31,27);(1,3,4,8)(2,5,6,7)(9,10,11,16,12,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,29)(30,32);(9,10,12,14)(11,15,13,16)(17,19,20,26)(18,22,23,31)(21,30,28,27)(24,32,29,25);(2,6)(5,7)(9,11,12,13)(10,15,14,16)(18,23)(22,31)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223945);