#180670

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

625 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31)(34,39)(35,42)(36,37)(41,45)(43,44)(46,47);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,12,14,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32)(33,34)(35,40)(36,45)(37,39)(38,41)(42,47)(43,46)(44,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(33,35,38,44)(34,40,41,48)(36,42,37,43)(39,46,45,47);(2,8)(3,10)(7,14)(9,16)(19,26)(22,30)(27,32)(29,31)(33,36,38,37)(34,39,41,45)(35,43,44,42)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=g,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, 87866);