#240645

order	512
nilpotency class	3
isoclinism	isoclinic class of #240639
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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)(3,11)(4,9)(5,7)(6,10)(8,12)(13,15)(14,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,30)(28,31)(29,32);(2,7)(3,12)(4,5)(8,16)(9,10)(11,15)(17,19,23,31)(18,24,26,32)(20,27,25,28)(21,30,22,29);(1,3,5,14,6,12,4,13)(2,8,9,16,10,11,7,15)(17,20,22,18,23,25,21,26)(19,28,30,24,31,27,29,32);(1,4,6,5)(2,9,10,7)(3,13,12,14)(8,16,11,15)(17,21,23,22)(18,25,26,20)(19,29,31,30)(24,27,32,28)

Polycyclic presentation:

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

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, 147841);