#269520

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59040	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176716);