#180410

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

612 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,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, 87606);