#268103

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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=h,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, 175299);