#234238

order	512
nilpotency class	3
isoclinism	isoclinic class of #234235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141434);