#305165

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efh,d^2=ehi,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=dh,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, 212361);