#322484

order	512
nilpotency class	3
isoclinism	isoclinic class of #322453
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229680);