#169720

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,7)(3,5,8,6)(10,15)(11,16)(12,13)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(9,10,14,11)(12,16,13,15)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(2,5)(3,8)(6,7)(9,11,14,10)(12,16,13,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(9,12,14,13)(10,15,11,16)(18,24)(22,28)(23,32)(30,31)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76916);