#233918

order	512
nilpotency class	3
isoclinism	isoclinic class of #233907
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

1300 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)(16,24)(22,37)(23,31)(25,32)(26,40)(27,34)(28,36)(29,35)(30,39)(33,38);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,33)(18,21)(19,32)(20,31)(22,28)(27,38)(29,39)(30,34)(35,37)(36,40);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,30)(18,36)(19,34)(20,35)(21,33)(23,39)(24,26)(25,38)(31,40)(32,37);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,32)(17,34,35,36)(22,26,38,39)(30,37,40,33)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi

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, 141114);