#201631

order	512
nilpotency class	3
isoclinism	isoclinic class of #201414
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1113 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108827);