#268670

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
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	#61868	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175866);