#332824

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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)(11,21)(12,26)(13,14)(18,33)(19,20)(22,31)(23,35)(24,25)(27,34)(36,40)(38,39);(1,3,5,4)(2,6,8,7)(9,10,15,20)(11,22,25,31)(12,27,26,33)(13,19,14,16)(17,21,32,24)(18,28,34,29)(23,37,39,30)(35,40,38,36);(2,7)(3,4)(6,8)(9,11)(10,17)(12,23)(13,24)(14,21)(15,25)(16,22)(18,30)(19,31)(20,32)(26,39)(27,36)(28,35)(29,38)(33,40)(34,37);(2,7)(3,4)(6,8)(9,12,15,26)(10,18,20,34)(11,23,25,39)(13,29,14,28)(16,27,19,33)(17,30,32,37)(21,35,24,38)(22,36,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240020);