#278584

order	512
nilpotency class	3
isoclinism	isoclinic class of #278563
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

511 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eh,c^2=i,d^2=e,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,d^c=dh,f^c=fi,g^c=ghi

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, 185780);