#168941

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2679	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,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, 76137);