#239258

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60676	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

898 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,29)(27,31)(30,32);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,19)(18,23)(20,29)(21,30)(22,31)(24,27)(25,28)(26,32);(17,20,22,21)(18,24,26,25)(19,29,31,30)(23,27,32,28)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146454);