#234435

order	512
nilpotency class	3
isoclinism	isoclinic class of #234422
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1371 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:

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

Polycyclic presentation:

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

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, 141631);