#322422

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eghi,d^2=eh,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, 229618);