#200016

order	512
nilpotency class	3
isoclinism	isoclinic class of #199851
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

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, 107212);