#199712

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 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:

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

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=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh

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, 106908);