#289119

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

613 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:

(1,2,4,9,5,8,13,7)(3,6,11,15,12,14,10,16)(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);(1,3,4,11,5,12,13,10)(2,6,9,15,8,14,7,16)(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);(2,7)(3,10)(4,13)(6,14)(8,9)(11,12)(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);(2,8)(6,14)(7,9)(15,16)(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=f,b^2=ef,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,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, 196315);