#190302

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=h,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, 97498);