#199532

order	512
nilpotency class	3
isoclinism	isoclinic class of #199528
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	#60965	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)(3,6,9,5)(4,11)(8,13)(12,16)(14,15)(17,18)(19,23,29,37)(20,30,32,24)(21,25)(22,26)(27,35)(28,36,44,48)(31,39,33,38)(34,40)(41,46)(42,45)(43,47);(1,3,5,2,7,9,6,10)(4,11,14,8,12,16,13,15)(17,19,22,29)(18,23)(20,27,32,42)(21,28,34,44)(24,35)(25,36)(26,37)(30,45)(31,41,33,43)(38,46)(39,47)(40,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,16)(10,15)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,39)(26,30)(28,43)(29,42)(31,34)(36,47)(37,45)(38,40)(41,44)(46,48);(4,12)(8,15)(11,16)(13,14)(17,21,22,34)(18,25,26,40)(19,28,29,44)(20,31,32,33)(23,36,37,48)(24,38,30,39)(27,41,42,43)(35,46,45,47)

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,d^c=dh,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, 106728);