#113305

order	512
nilpotency class	3
isoclinism	isoclinic class of #113302
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

813 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,f^c=fh,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, 20501);