#237363

order	512
nilpotency class	3
isoclinism	isoclinic class of #237345
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	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1302 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(18,23)(19,27)(20,21)(25,26)(28,32)(29,30);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,18)(19,28)(20,26)(21,23)(22,25)(24,27)(29,31)(30,32);(1,3,6,12)(2,8,10,16)(4,13,5,11)(7,14,9,15)(17,19,22,30)(18,24,25,31)(20,27,21,29)(23,28,26,32);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(18,25)(23,26)(24,31)(28,32)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,f^2=i,g^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,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

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, 144559);