#179632

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

703 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86828);