#488972

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61088	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

909 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

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, 396168);