#113591

order	512
nilpotency class	3
isoclinism	isoclinic class of #113584
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

862 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20787);