#263492

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,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, 170688);