#110656

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

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, 17852);