#280976

order	512
nilpotency class	3
isoclinism	isoclinic class of #280955
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28240	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=ei,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,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, 188172);