#290054

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 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)(19,30)(20,25)(21,36)(22,28)(23,29)(24,32)(26,37)(27,34)(31,45)(33,40)(35,39)(38,43)(41,47)(42,46)(44,48);(1,3,4,13,6,14,5,12)(2,7,9,15,11,16,10,8)(17,19,22,32,23,20,34,33)(18,24,27,30,29,40,28,25)(21,31,37,46,38,35,48,47)(26,39,36,42,44,45,43,41);(1,3,6,14)(2,8,11,15)(4,12,5,13)(7,9,16,10)(17,20)(18,25)(19,23)(21,31)(22,33)(24,27)(26,42)(28,40)(29,30)(32,34)(35,38)(36,45)(37,46)(39,43)(41,44)(47,48);(1,4,6,5)(2,9,11,10)(3,12,14,13)(7,8,16,15)(17,21,34,48,23,38,22,37)(18,26,28,43,29,44,27,36)(19,31,33,47,20,35,32,46)(24,39,25,41,40,45,30,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=ehi,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, 197250);