#172343

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79539);