#190303

order	512
nilpotency class	3
isoclinism	isoclinic class of #190234
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	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18,22,28,23,26,21,27)(19,29,32,41,31,24,36,39)(20,33,34,40,35,42,37,25)(30,44,47,43,45,48,46,38);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19)(18,24)(20,30)(21,32)(22,36)(23,31)(25,38)(26,29)(27,39)(28,41)(33,48)(34,46)(35,45)(37,47)(40,43)(42,44);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25,26,40)(19,30)(21,34)(22,37)(23,35)(24,38,29,43)(27,33,28,42)(31,45)(32,46)(36,47)(39,48,41,44);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(19,31)(24,29)(30,45)(32,36)(38,43)(39,41)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,f^c=fi,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, 97499);