#180754

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

662 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,25)(19,26,29,32)(20,24,21,30)(23,27,31,28)(33,34,38,40)(35,43,44,48)(36,42,37,41)(39,46,47,45);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,48)(42,43);(1,3,5,10)(2,7,8,15)(4,11,12,16)(6,13,9,14)(20,21)(24,30)(26,32)(27,28)(33,35)(34,39)(36,45)(37,46)(38,44)(40,47)(41,43)(42,48);(2,8)(3,10)(6,9)(11,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)(34,40)(35,44)(41,42)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87950);