#236759

order	512
nilpotency class	3
isoclinism	isoclinic class of #236755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143955);