#252012

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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:

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

Polycyclic presentation:

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

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, 159208);