#263489

order	512
nilpotency class	3
isoclinism	isoclinic class of #263459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18687	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(18,23)(19,26)(20,21)(24,27)(25,30)(28,29);(1,3,6,12)(2,7,10,16)(4,11,5,13)(8,15,9,14)(17,18,21,30,22,25,20,23)(19,24,28,32,29,27,26,31);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,18)(19,27)(20,23)(21,30)(22,25)(24,29)(26,32)(28,31);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,19,20,26,22,29,21,28)(18,24,23,31,25,27,30,32)

Polycyclic presentation:

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

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, 170685);