#294790

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

919 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)(9,10,15,11)(12,14,16,13)(17,18,21,25)(19,26,22,23)(20,28,29,31)(24,27,32,30);(2,5)(6,7)(9,11,15,10)(12,13,16,14)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,32,28);(9,12,14,11,15,16,13,10)(17,20,22,30,21,29,19,27)(18,24,26,28,25,32,23,31);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,12,11,16)(18,25)(23,26)(24,32)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=eh,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, 201986);