#240921

order	512
nilpotency class	3
isoclinism	isoclinic class of #240913
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60959	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,g^b=gh,d^c=di,f^c=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, 148117);