#169116

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76312);