#179227

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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

661 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26)(33,34)(35,43)(36,41)(37,40)(38,42)(39,45)(44,48)(46,47);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25)(33,35,38,46)(34,39,42,48)(36,44,37,45)(40,47,41,43);(1,3,6,12)(2,8,10,16)(4,13,5,11)(7,14,9,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,32)(27,31)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(18,23)(22,31)(24,28)(25,32)(33,36,38,37)(34,41,42,40)(35,44,46,45)(39,43,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,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, 86423);