#180547

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=ghi,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, 87743);