#263974

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	#121	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

(1,2,7,11)(3,8,13,4)(5,16,6,10)(9,15,12,14)(17,18)(19,31)(20,36)(21,39)(22,29)(23,28)(24,30)(25,34)(26,38)(27,32)(33,48)(35,47)(37,46)(40,45)(41,43)(42,44);(1,3)(2,8)(4,11)(5,12)(6,9)(7,13)(10,15)(14,16)(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);(1,4,7,8)(2,9,11,12)(3,10,13,16)(5,15,6,14)(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=hi,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, 171170);