#249732

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156928);