#279202

order	512
nilpotency class	3
isoclinism	isoclinic class of #279179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186398);