#237471

order	512
nilpotency class	3
isoclinism	isoclinic class of #237463
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1226 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:

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

Polycyclic presentation:

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

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, 144667);