#198765

order	512
nilpotency class	3
isoclinism	isoclinic class of #198528
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1073 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105961);