#180546

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

609 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=ghi,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, 87742);