#273300

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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

830 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:

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

Polycyclic presentation:

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

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, 180496);