#273210

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
prime radical	#62166	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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)(7,15,16,10)(17,18,23,27)(19,29,25,31)(20,30,21,24)(22,28,32,26);(1,2)(3,7)(4,8)(5,13)(6,9)(10,11)(12,16)(14,15)(17,19,23,25)(18,21,27,20)(22,24,32,30)(26,31,28,29);(1,3,6,12)(2,7,9,16)(4,10,13,15)(5,14,8,11)(17,20)(18,24)(19,26)(21,23)(22,31)(25,28)(27,30)(29,32);(1,4,6,13)(2,8,9,5)(3,10,12,15)(7,11,16,14)(17,21)(18,25)(19,27)(20,23)(22,31)(24,28)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=hi,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, 180406);