#203069

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110265);