#263971

order	512
nilpotency class	3
isoclinism	isoclinic class of #263963
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	#62437	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18,24,30)(19,31,35,47)(20,36,33,48)(21,39,40,45)(22,29,23,28)(25,41,43,34)(26,42,44,38)(27,37,46,32);(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,19,22,34,24,35,23,41)(18,25,28,31,30,43,29,47)(20,37,38,21,33,32,42,40)(26,27,36,39,44,46,48,45);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,20,24,33)(18,26,30,44)(19,32,35,37)(21,34,40,41)(22,38,23,42)(25,27,43,46)(28,36,29,48)(31,39,47,45);(17,21,23,37,24,40,22,32)(18,27,29,45,30,46,28,39)(19,33,41,38,35,20,34,42)(25,36,47,26,43,48,31,44)

Polycyclic presentation:

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

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, 171167);