#179012

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,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, 86208);