#180506

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

703 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=i,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, 87702);