#237530

order	512
nilpotency class	3
isoclinism	isoclinic class of #237527
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

1290 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:

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

Polycyclic presentation:

2^9|a^2=h,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,d^c=dhi,e^c=ei,f^c=fhi,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, 144726);