#273822

order	512
nilpotency class	3
isoclinism	isoclinic class of #273819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181018);