#190159

order	512
nilpotency class	3
isoclinism	isoclinic class of #190124
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

926 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,27,21,26)(19,29,31,42,32,24,37,39)(20,33,36,40,35,25,34,41)(30,38,44,47,45,43,46,48);(4,12)(7,14)(10,16)(13,15)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,39)(27,29)(28,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,25,27,33)(19,30)(21,34)(22,36)(23,35)(24,38,29,43)(26,40,28,41)(31,44)(32,45)(37,46)(39,47,42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,23,22)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,33,41)(30,44,45,46)(38,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97355);