#180533

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

624 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)(3,7)(4,12)(5,9)(6,11)(8,14)(10,16)(13,15)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(33,34,38,35)(36,43,44,48)(37,42,47,40)(39,46,41,45);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,19)(18,23)(20,29)(21,28)(22,30)(24,27)(25,31)(26,32)(33,35,38,34)(36,41,44,39)(37,40,47,42)(43,46,48,45);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(20,21)(24,25)(27,31)(28,29)(33,36)(34,39)(35,41)(37,46)(38,44)(40,48)(42,43)(45,47);(3,10)(6,15)(7,16)(11,13)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)(34,35)(39,41)(40,42)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 87729);