#199646

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	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

1014 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,g^2=i,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, 106842);