#302480

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=f,d^2=e,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=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209676);