#332688

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
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

645 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239884);