#273263

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#3561	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 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)(3,10,12,15)(4,8)(5,9)(6,11,7,13)(14,16)(17,18)(19,29)(20,26)(21,32)(22,39)(23,27)(24,28)(25,31)(30,46)(33,38)(34,41)(35,48)(36,44)(37,45)(40,43)(42,47);(1,3)(2,6)(4,13)(5,12)(7,9)(8,15)(10,16)(11,14)(17,19,24,33)(18,25,28,21)(20,34,37,48)(22,32,23,31)(26,42,45,46)(27,38,39,29)(30,40,47,36)(35,44,41,43);(4,14)(8,16)(10,15)(11,13)(17,20,24,37)(18,26,28,45)(19,30,33,47)(21,35,25,41)(22,36,23,40)(27,43,39,44)(29,46,38,42)(31,34,32,48);(1,3)(2,7)(4,13)(5,12)(6,9)(8,10)(11,14)(15,16)(17,21)(18,19)(20,35)(22,29)(23,38)(24,25)(26,30)(27,32)(28,33)(31,39)(34,44)(36,46)(37,41)(40,42)(43,48)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180459);