#240576

order	512
nilpotency class	3
isoclinism	isoclinic class of #240563
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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)(5,13)(6,10)(8,11)(12,16)(14,15)(17,18,22,25)(19,26,29,32)(20,30,24,21)(23,27,31,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(17,19,22,29)(18,23,25,31)(20,26,24,32)(21,27,30,28);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,20,21,18,22,24,30,25)(19,26,28,31,29,32,27,23);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,8,16,15)(17,21,22,30)(18,20,25,24)(19,27,29,28)(23,26,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147772);