#200756

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1076 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)(3,7,12,16)(4,10)(5,8)(6,9)(11,14,13,15)(17,18)(19,29)(20,34,37,41)(21,26)(22,27)(23,28)(24,32)(25,36,42,35)(30,44,47,38)(31,40)(33,39)(43,46,48,45);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,14,10,15)(17,19,21,32,23,33,22,31)(18,24,27,29,28,40,26,39)(20,30,36,45,37,47,35,46)(25,38,41,43,42,44,34,48);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,34)(27,41)(28,42)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44);(1,4,6,5)(2,8,9,10)(3,11,12,13)(7,15,16,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,41,42,34)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

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, 107952);