#180309

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

611 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(20,27)(22,28)(23,30)(25,31)(33,34,39,35)(36,44,46,48)(37,42,38,41)(40,47,43,45);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32)(33,35,39,34)(36,43,46,40)(37,41,38,42)(44,47,48,45);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,15)(10,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32)(33,36,39,46)(34,40,35,43)(37,45,38,47)(41,44,42,48);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)(33,37,39,38)(34,41,35,42)(36,45,46,47)(40,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,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, 87505);