#274392

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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)(9,10,14,15,13,12,11,16)(17,18,20,25,21,26,22,24)(19,27,29,32,30,23,28,31);(1,3,4,8)(2,5,6,7)(9,11,13,14)(10,15,12,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,31)(26,27);(9,12,13,10)(11,16,14,15)(18,24)(19,28)(20,22)(23,27)(25,26)(29,30);(2,6)(5,7)(17,20,21,22)(18,25,26,24)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ei,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181588);