#190067

order	512
nilpotency class	3
isoclinism	isoclinic class of #190065
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	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

817 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)(3,11,14,7)(4,9)(5,8)(6,10)(12,15,13,16)(17,18)(19,27,29,23)(20,31,33,46)(21,26)(22,25)(24,32,37,47)(28,41)(30,40,44,36)(34,38)(35,43)(39,48)(42,45);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,27)(26,36)(31,45)(32,43)(33,42)(34,44)(37,39)(38,40)(41,46)(47,48);(2,8,10,9)(7,15,11,16)(17,20)(18,24,25,37)(19,28)(21,32)(22,33)(23,35,27,39)(26,31,38,46)(29,42)(30,43)(34,47)(36,45,40,41)(44,48);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,16,11,15)(19,29)(23,27)(28,42)(30,44)(35,39)(36,40)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,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, 97263);