#180538

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

589 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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, 87734);