#236922

order	512
nilpotency class	3
isoclinism	isoclinic class of #10144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1206 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144118);