#278540

order	512
nilpotency class	3
isoclinism	isoclinic class of #278523
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	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

514 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:

(1,2,5,16,7,11,6,10)(3,12,13,8,4,15,14,9)(17,18)(19,29)(20,33)(21,37)(22,28)(23,27)(24,31)(25,32)(26,34)(30,44)(35,47)(36,42)(38,40)(39,46)(41,48)(43,45);(1,3,6,14,7,4,5,13)(2,8,16,15,11,9,10,12)(17,19,23,38,22,32,21,31)(18,24,37,25,28,40,27,29)(20,30,36,48,35,46,34,45)(26,39,47,41,42,44,33,43);(1,2,7,11)(3,8,4,9)(5,10,6,16)(12,14,15,13)(17,19)(18,25)(20,30)(21,31)(22,32)(23,38)(24,27)(26,41)(28,29)(33,39)(34,45)(35,46)(36,48)(37,40)(42,43)(44,47);(1,4,6,13,7,3,5,14)(2,9,16,12,11,8,10,15)(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=fh,b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185736);