#237409

order	512
nilpotency class	3
isoclinism	isoclinic class of #237395
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	#60946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1224 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)(19,29)(20,30)(21,26)(22,28)(23,27)(24,25)(31,32);(1,2)(3,11)(4,13)(5,7)(6,9)(8,12)(10,15)(14,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,25)(26,29)(27,32);(1,3,6,12)(2,8,9,11)(4,10,5,14)(7,15,13,16)(17,20,22,24)(18,25,28,30)(19,26,31,27)(21,29,23,32);(3,12)(8,11)(10,14)(15,16)(17,21,22,23)(18,26,28,27)(19,30,31,25)(20,32,24,29)

Polycyclic presentation:

2^9|c^2=h,d^2=i,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,e^b=eh,g^b=gi,d^c=dh,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, 144605);