#278660

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
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

498 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,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, 185856);