#119305

order	512
nilpotency class	3
isoclinism	isoclinic class of #119300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=eh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26501);