#332664

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

640 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:

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

Polycyclic presentation:

2^9|a^2=i,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=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, 239860);