#273221

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
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

891 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=i,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,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180417);