#199524

order	512
nilpotency class	3
isoclinism	isoclinic class of #199522
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	C₂³×C₄	C₂⁴
prime radical	#60951	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1046 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)(8,12)(13,16)(14,15)(17,18)(19,23,29,37)(20,30,32,24)(21,25)(22,26)(27,35)(28,36,33,39)(31,44,46,38)(34,40)(41,47)(42,43)(45,48);(1,3,6,2,7,9,5,10)(4,11,13,8,14,15,12,16)(17,19,22,29)(18,23)(20,27,32,42)(21,33,34,28)(24,35)(25,39)(26,37)(30,43)(31,45,46,41)(36,40)(38,48)(44,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,41)(29,42)(33,45)(34,46)(36,47)(37,43)(39,48)(40,44);(1,5,7,6)(2,9,10,3)(4,12,14,13)(8,15,16,11)(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,41,42,45)(35,47,43,48)

Polycyclic presentation:

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

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, 106720);