#287526

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

593 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:

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

Polycyclic presentation:

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

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, 194722);