#185606

order	512
nilpotency class	3
isoclinism	isoclinic class of #185517
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	C₂×#540	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2035 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:

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

Polycyclic presentation:

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

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, 92802);