#273913

order	512
nilpotency class	3
isoclinism	isoclinic class of #273819
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=i,e^2=i,f^2=hi,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=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181109);