#290073

order	512
nilpotency class	3
isoclinism	isoclinic class of #290011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197269);