#200604

order	512
nilpotency class	3
isoclinism	isoclinic class of #200430
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,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, 107800);