#173846

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
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

654 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,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, 81042);