#180518

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 87714);