#178888

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

675 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86084);