#113359

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,18)(19,27,29,23)(20,31,33,24)(21,26)(22,25)(28,45)(30,43,35,39)(32,48)(34,44,36,41)(37,42)(38,46)(40,47);(1,2,6,10)(3,11,14,15)(4,9,5,7)(8,12,16,13)(17,19,22,29)(18,23,25,27)(20,32,33,47)(21,35,37,30)(24,40,31,48)(26,43,42,39)(28,36,46,34)(38,44,45,41);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20,22,33)(18,24,25,31)(19,28,29,46)(21,36,37,34)(23,38,27,45)(26,44,42,41)(30,47,35,32)(39,48,43,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^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, 20555);