#170468

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
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	#37192	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

859 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,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, 77664);