#201854

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109050);