#206487

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	#60693	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113683);