#256693

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,28,25)(20,29,30,32)(21,26,22,31);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(14,16)(17,19)(18,24)(20,21)(22,30)(23,28)(25,27)(26,29)(31,32);(1,4,5,3)(2,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,20,23,30)(18,25,27,24)(19,21,28,22)(26,32,31,29);(2,7)(6,8)(10,14)(13,16)(17,21,23,22)(18,26,27,31)(19,20,28,30)(24,29,25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,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, 163889);