#237323

order	512
nilpotency class	3
isoclinism	isoclinic class of #237305
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	C₂×#3251	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1198 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(18,23)(19,27)(20,21)(25,26)(28,31)(29,30);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,18,22,25)(19,28,29,31)(20,26,21,23)(24,30,32,27);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,25)(19,29)(23,26)(27,30)

Polycyclic presentation:

2^9|b^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=dh,e^c=ei,f^c=fi,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, 144519);