#273697

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	#62235	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 180893);