#273655

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
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	#61220	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,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, 180851);