#199258

order	512
nilpotency class	3
isoclinism	isoclinic class of #199251
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,23,28,38)(20,30)(21,33)(22,25)(24,32)(26,35)(27,41,42,48)(29,45,44,37)(31,40)(34,39)(36,46,47,43);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,19)(18,23,25,38)(20,27)(21,29)(22,28)(24,36,39,47)(26,37,33,45)(30,41,40,48)(31,42)(32,43)(34,46)(35,44);(1,2)(3,7)(4,14)(5,8)(6,9)(10,11)(12,16)(13,15)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,39)(26,40)(28,42)(29,46)(30,33)(32,35)(37,48)(38,47)(41,45)(43,44);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(18,25)(19,28)(20,31)(26,33)(29,44)(32,34)(36,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,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, 106454);