#273298

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	#62017	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

817 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)(4,14,15,7)(5,10)(6,8)(13,16)(17,18)(19,26,21,24)(20,25)(22,28)(23,27)(29,38,32,41)(30,44,35,39)(31,40,47,48)(33,43)(34,42)(36,46)(37,45);(1,3)(2,7)(4,13)(5,15)(6,12)(8,14)(9,10)(11,16)(17,19,23,21)(18,24,27,26)(20,31,33,47)(22,35,37,30)(25,40,43,48)(28,44,45,39)(29,36,32,34)(38,46,41,42);(2,8)(3,12)(5,13)(9,11)(17,20,23,33)(18,25,27,43)(19,29,21,32)(22,36,37,34)(24,38,26,41)(28,46,45,42)(30,47,35,31)(39,48,44,40);(1,4,6,15)(2,9,8,11)(3,13,12,5)(7,10,14,16)(17,21,23,19)(18,26,27,24)(20,32,33,29)(22,35,37,30)(25,41,43,38)(28,44,45,39)(31,34,47,36)(40,42,48,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,d^2=hi,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, 180494);