#249804

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1346 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 157000);