#278629

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
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

495 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,13)(14,16)(17,18,23,28,24,29,22,38)(19,27,21,37,32,44,31,25)(20,33,35,43,36,42,39,26)(30,40,45,47,46,48,34,41);(1,2,5,15,6,9,4,8)(3,11,13,10,14,7,12,16)(17,19,23,21,24,32,22,31)(18,25,38,44,29,37,28,27)(20,30,39,34,36,46,35,45)(26,40,33,47,43,48,42,41);(1,2)(3,7)(4,8)(5,15)(6,9)(10,12)(11,14)(13,16)(17,20,24,36)(18,26,29,43)(19,30,32,46)(21,34,31,45)(22,35,23,39)(25,40,37,48)(27,41,44,47)(28,42,38,33);(1,3,4,12,6,14,5,13)(2,7,8,10,9,11,15,16)(17,21,23,32,24,31,22,19)(18,27,38,25,29,44,28,37)(20,34,39,46,36,45,35,30)(26,41,33,40,43,47,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,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,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185825);