#322408

order	512
nilpotency class	3
isoclinism	isoclinic class of #322377
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

640 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eghi,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,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, 229604);