#180567

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

636 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)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,47)(46,48);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19)(18,23)(20,29)(21,28)(22,30)(24,27)(25,31)(26,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(20,21)(24,25)(27,31)(28,29)(33,35)(34,39)(36,45)(37,44)(38,46)(40,48)(41,43)(42,47);(2,8)(3,9)(6,12)(10,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,43,47,48)

Polycyclic presentation:

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

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, 87763);