#249801

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

1325 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)(11,20)(12,25)(13,28)(14,18)(15,19)(16,21)(17,26)(22,31)(23,30)(24,32)(27,34)(29,33)(35,37)(36,39)(38,40);(3,7)(5,8)(9,11)(10,16)(12,22)(13,24)(14,21)(15,23)(17,31)(18,30)(19,32)(20,28)(25,35)(26,36)(27,38)(29,37)(33,40)(34,39);(1,2)(3,8)(4,6)(5,7)(11,21)(13,14)(16,30)(18,28)(20,32)(22,36)(23,24)(25,34)(26,29)(31,39)(35,40)(37,38);(3,7)(5,8)(9,12,15,27)(10,17,19,33)(11,22,23,38)(13,29,14,26)(16,31,32,40)(18,34,28,25)(20,35,30,39)(21,36,24,37)

Polycyclic presentation:

2^9|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, 156997);