#113360

order	512
nilpotency class	3
isoclinism	isoclinic class of #113356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60942	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20556);