#273749

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62090	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 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)(3,11,12,9)(4,7,14,16)(5,10)(6,8)(13,15)(17,18,22,39,24,21,23,27)(19,28,32,30,33,42,31,25)(20,34,40,26,38,44,37,36)(29,46,35,45,48,41,47,43);(1,3,6,12)(2,7,8,16)(4,13,14,5)(9,15,11,10)(17,19)(18,25)(20,35)(21,30)(22,32)(23,31)(24,33)(26,43)(27,42)(28,39)(29,37)(34,41)(36,45)(38,47)(40,48)(44,46);(2,8)(3,12)(5,13)(9,11)(17,20,24,38)(18,26,21,36)(19,29,33,48)(22,37,23,40)(25,41,30,46)(27,44,39,34)(28,45,42,43)(31,35,32,47);(1,4)(2,9)(3,13)(5,12)(6,14)(7,15)(8,11)(10,16)(17,21)(18,24)(19,30)(20,36)(22,27)(23,39)(25,33)(26,38)(28,31)(29,46)(32,42)(34,40)(35,45)(37,44)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

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, 180945);