#180318

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

592 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)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(34,39)(35,42)(36,37)(41,44)(43,45)(46,48);(1,2,6,10)(3,8,13,16)(4,7,5,9)(11,15,12,14)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27)(33,34)(35,40)(36,44)(37,39)(38,41)(42,48)(43,47)(45,46);(1,3,6,13)(2,8,10,16)(4,12,5,11)(7,14,9,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)(33,35)(34,40)(36,42)(37,45)(38,43)(39,46)(41,47)(44,48);(1,4,6,5)(2,9,10,7)(3,11,13,12)(8,14,16,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)(35,43)(40,47)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,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, 87514);