#248411

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

1176 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)(18,23)(19,26)(20,21)(25,29)(28,30)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,18,22,25)(19,24,28,27)(20,29,21,23)(26,32,30,31);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,19,22,28)(18,24,25,27)(20,26,21,30)(23,31,29,32);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,18,22,25)(19,27,28,24)(20,29,21,23)(26,31,30,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,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, 155607);