#180722

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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

566 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)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(35,41)(36,43)(38,45)(40,47)(42,44)(46,48);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27)(33,34)(35,38)(36,40)(37,39)(41,45)(42,46)(43,47)(44,48);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29)(33,35,37,42)(34,38,39,46)(36,44,43,41)(40,48,47,45);(2,4)(7,10)(8,13)(9,16)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31)(34,39)(38,46)(40,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dhi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87918);