#488859

order	512
nilpotency class	4
isoclinism	isoclinic class of #488719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3259	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396055);