#110655

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
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

594 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)(3,10,13,14)(4,6,11,8)(7,15,16,12)(17,18,21,28,23,26,22,27)(19,24,31,39,30,40,36,38)(20,32,34,25,33,41,35,42)(29,43,44,37,45,46,47,48);(2,6,9,8)(7,14,16,10)(17,19,22,31,23,30,21,36)(18,24,28,38,26,40,27,39)(20,29,35,44,33,45,34,47)(25,37,41,43,42,48,32,46);(1,3,11,15,5,13,4,12)(2,7,6,14,9,16,8,10)(17,20,23,33)(18,25)(19,29,30,45)(21,34,22,35)(24,37)(26,42)(27,32)(28,41)(31,44,36,47)(38,43)(39,46)(40,48)

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,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17851);