#332332

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 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)(3,5)(4,7)(6,8)(9,10)(11,20)(12,17)(13,25)(14,28)(15,19)(16,21)(18,24)(22,39)(23,33)(26,35)(27,34)(29,32)(30,36)(31,40)(37,38);(9,11)(10,16)(12,21)(13,26)(14,29)(15,23)(17,33)(18,35)(19,32)(20,28)(22,24)(25,31)(27,38)(30,40)(34,39)(36,37);(2,5)(3,8)(6,7)(9,12,15,14)(10,17,19,28)(11,21,23,29)(13,24,27,30)(16,20,32,33)(18,34,36,25)(22,38,40,26)(31,37,39,35);(2,5)(3,8)(6,7)(9,13,15,27)(10,18,19,36)(11,22,23,40)(12,24,14,30)(16,31,32,39)(17,34,28,25)(20,37,33,35)(21,38,29,26)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fi,e^d=eh,f^d=fi

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, 239528);