#199490

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,f^c=fi

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, 106686);