#112552

order	512
nilpotency class	3
isoclinism	isoclinic class of #10326
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19748);