#168039

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37219	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75235);