#268100

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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	C₂×#553	C₂²
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:

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

Polycyclic presentation:

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

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, 175296);