#322415

order	512
nilpotency class	3
isoclinism	isoclinic class of #322409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19676	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=ghi

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, 229611);