#237127

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1278 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

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, 144323);