#199917

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	#60917	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,g^2=h,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, 107113);