#237318

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	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,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, 144514);