#240572

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18)(19,27)(20,22)(21,25)(23,26)(24,29)(28,31)(30,32);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,19,23,31)(18,24,26,32)(20,27,25,28)(21,29,22,30);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,20,22,18,23,25,21,26)(19,28,29,24,31,27,30,32);(17,21,23,22)(18,25,26,20)(19,29,31,30)(24,28,32,27)

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,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, 147768);