#249802

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

1365 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)(14,19)(15,20)(16,24)(17,29)(18,28)(22,38)(23,33)(25,35)(26,34)(30,37)(32,39)(36,40);(3,7)(5,8)(9,11)(10,16)(12,23)(13,25)(14,24)(15,26)(17,36)(18,33)(19,34)(20,35)(21,31)(22,28)(27,32)(29,39)(30,40)(37,38);(1,2,4,6)(3,8,7,5)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,32)(18,20)(19,37)(21,33)(23,24)(25,40)(26,39)(27,31)(34,36)(35,38);(3,7)(5,8)(9,12)(10,18)(11,23)(13,29)(14,28)(15,30)(16,33)(17,20)(19,37)(21,32)(22,24)(25,39)(26,40)(27,31)(34,38)(35,36)

Polycyclic presentation:

2^9|c^2=i,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, 156998);