#273685

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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 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)(3,12)(4,9)(5,8)(6,15)(7,11)(10,13)(14,16)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,45,46,38)(29,43,36,39)(32,48,47,40)(34,44)(35,42)(37,41);(1,3,7,13)(2,8,11,16)(4,5,6,14)(9,10,15,12)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,40,31,48)(26,43,42,39)(28,37,46,34)(38,44,45,41);(1,4,7,6)(2,9,11,15)(3,5,13,14)(8,10,16,12)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,35)(39,48)(40,43)(41,42);(1,5,7,14)(2,10,11,12)(3,4,13,6)(8,9,16,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,38)(25,27)(26,39)(31,45)(32,34)(33,46)(35,36)(37,47)(40,41)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=i,d^2=i,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, 180881);