#240840

order	512
nilpotency class	3
isoclinism	isoclinic class of #240831
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

864 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,21)(22,25)(23,26)(24,29)(28,30)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,23,31)(18,24,26,28)(20,27,25,32)(21,30,22,29);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20,21,18,23,25,22,26)(19,28,29,32,31,24,30,27);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,23,22)(18,20,26,25)(19,29,31,30)(24,32,28,27)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=h,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,g^b=gh,d^c=di,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, 148036);