#113650

order	512
nilpotency class	3
isoclinism	isoclinic class of #113643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

946 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:

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

Polycyclic presentation:

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

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, 20846);