#199414

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1027 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:

(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18,22,26,23,27,21,35)(19,24,31,38,30,40,29,39)(20,32,33,25,34,42,36,41)(28,43,44,48,45,47,46,37);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,22,29,23,30,21,31)(18,24,35,38,27,40,26,39)(20,28,33,46,34,45,36,44)(25,37,32,43,41,48,42,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,25)(19,28)(21,36)(22,33)(23,34)(24,37)(26,42)(27,41)(29,46)(30,45)(31,44)(32,35)(38,43)(39,47)(40,48);(2,8)(3,9)(6,13)(10,15)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,40,39)(25,32,41,42)(28,44,45,46)(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=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, 106610);