#118211

order	512
nilpotency class	3
isoclinism	isoclinic class of #118207
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fhi,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, 25407);