#248418

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
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

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155614);