#235393

order	512
nilpotency class	3
isoclinism	isoclinic class of #235381
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

1254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142589);