#252893

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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)(9,10,12,14)(11,16,13,15)(18,23)(19,27)(20,22)(24,28)(29,30)(31,32);(1,3,4,8)(2,5,6,7)(10,14)(15,16)(17,18,21,23)(19,28,29,32)(20,25,22,26)(24,27,31,30);(10,15)(11,13)(14,16)(17,19,21,29)(18,24,23,31)(20,30,22,27)(25,32,26,28);(2,6)(5,7)(9,11,12,13)(10,15,14,16)(17,20,21,22)(18,25,23,26)(19,27,29,30)(24,28,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160089);