#185446

order	512
nilpotency class	3
isoclinism	isoclinic class of #185285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh

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, 92642);