#250794

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
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

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 157990);