#288651

order	512
nilpotency class	3
isoclinism	isoclinic class of #288643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,13,16,15,20,14,19)(11,21,22,17,25,32,24,33)(12,26,28,18,27,30,29,34)(23,31,36,39,38,40,37,35);(1,2,4,7)(3,5,8,6)(10,16,20,19)(11,22,25,24)(12,27)(18,26,34,30)(23,37,38,36)(28,29)(31,40)(35,39);(2,5)(3,8)(6,7)(9,11)(10,17)(12,23)(13,24)(14,22)(15,25)(16,21)(18,31)(19,32)(20,33)(26,39)(27,38)(28,37)(29,36)(30,35)(34,40);(9,12,14,29,15,27,13,28)(10,18,16,30,20,34,19,26)(11,23,22,36,25,38,24,37)(17,31,21,35,33,40,32,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=efi,d^2=eh,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195847);