#198178

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1030 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)(3,6)(4,12,13,10)(5,11)(7,9)(8,15,16,14)(17,18)(19,23,29,38)(20,24)(21,32)(22,26)(25,35)(27,36,44,47)(28,41,33,37)(30,42)(31,40)(34,39)(43,48,45,46);(1,3,6,2,7,11,5,9)(4,10,14,8,13,12,15,16)(17,19,22,29)(18,23)(20,27,31,44)(21,33,35,28)(24,36)(25,41)(26,38)(30,45,34,43)(32,37)(39,48)(40,47)(42,46);(1,4,7,13)(2,8,9,16)(3,10,11,12)(5,15,6,14)(17,20,22,31)(18,24,26,40)(19,27,29,44)(21,34,35,30)(23,36,38,47)(25,42,32,39)(28,45,33,43)(37,48,41,46);(4,13)(8,16)(10,12)(14,15)(17,21,22,35)(18,25,26,32)(19,28,29,33)(20,30,31,34)(23,37,38,41)(24,39,40,42)(27,43,44,45)(36,46,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105374);