#239116

order	512
nilpotency class	3
isoclinism	isoclinic class of #233245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146312);