#169114

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

734 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27)(33,34,37,41)(35,38,43,47)(36,39,44,40)(42,45,48,46);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25)(33,35,37,43)(34,38,41,47)(36,42,44,48)(39,45,40,46);(20,30)(24,28)(25,32)(27,29)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,23)(19,26)(24,28)(27,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,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, 76310);