#115514

order	512
nilpotency class	3
isoclinism	isoclinic class of #10104
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

805 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)(3,10,12,6)(4,9)(5,8)(7,11,15,13)(14,16)(17,18,21,23,20,24,28,30)(19,22,27,31,26,32,29,25);(1,3,5,12)(2,6,8,10)(4,13,14,11)(7,9,15,16)(17,19,20,26)(18,22,24,32)(21,29,28,27)(23,25,30,31);(1,3)(2,7)(4,13)(5,12)(6,9)(8,15)(10,16)(11,14)(17,18)(19,25)(20,24)(21,30)(22,29)(23,28)(26,31)(27,32)

Polycyclic presentation:

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

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, 22710);