#244591

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
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

1053 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:

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=i,g^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=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

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, 151787);