#317029

order	512
nilpotency class	3
isoclinism	isoclinic class of #317015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#9996	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224225);