#348653

order	512
nilpotency class	3
isoclinism	isoclinic class of #348090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3531	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255849);