#258610

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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)(9,10,13,11)(12,15,14,16)(17,18,21,25)(19,26,22,23)(20,28,29,32)(24,30,31,27);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,28,31,32);(10,15)(11,16)(12,14)(17,20,21,29)(18,24,25,31)(19,27,22,30)(23,28,26,32);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,11,15)(18,25)(23,26)(24,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,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,d^c=di,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, 165806);