#199480

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
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

1136 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)(19,24,31,41)(20,33)(21,28)(22,26)(23,27)(25,35)(29,45,34,48)(30,40,32,39)(36,44)(37,43)(38,47,42,46);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19)(18,24,27,41)(20,34)(21,30)(22,32)(23,31)(25,42,43,38)(26,39,28,40)(29,36)(33,48,44,45)(35,47)(37,46);(3,11)(7,15)(10,16)(12,13)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,38)(26,33)(27,43)(28,44)(30,46)(31,34)(32,47)(39,45)(40,48)(41,42);(17,21,23,22)(18,26,27,28)(19,30,31,32)(20,35,36,37)(24,39,41,40)(25,33,43,44)(29,46,34,47)(38,45,42,48)

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,c^b=ch,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, 106676);