#258430

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

855 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)(4,5)(6,8)(9,10,12,15)(11,16,13,14)(17,18)(19,23)(20,22)(21,24)(25,32)(26,29)(27,30)(28,31);(2,5)(6,7)(9,11,12,13)(10,14,15,16)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(10,14)(11,13)(15,16)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(18,22)(23,29)(24,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165626);