#273713

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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	#62242	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,e^2=i,f^2=h,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=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180909);