#199520

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60951	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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)(3,6,9,5)(4,11)(8,12)(13,15)(14,16)(17,18)(19,23,29,36)(20,30,32,24)(21,25)(22,26)(27,34)(28,35,41,46)(31,43,44,37)(33,38)(39,45)(40,42)(47,48);(1,3,5,2,7,9,6,10)(4,11,13,8,14,16,12,15)(17,19,22,29)(18,23)(20,27,32,40)(21,28,33,41)(24,34)(25,35)(26,36)(30,42)(31,39,44,47)(37,45)(38,46)(43,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,37)(26,30)(28,39)(29,40)(33,44)(35,45)(36,42)(38,43)(41,47)(46,48);(17,21,22,33)(18,25,26,38)(19,28,29,41)(20,31,32,44)(23,35,36,46)(24,37,30,43)(27,39,40,47)(34,45,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 106716);