#193000

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,22,27,23,28,21,26)(19,29,32,24,33,38,31,39)(20,25,34,40,36,41,35,42)(30,43,44,37,45,47,46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,22,32,23,33,21,31)(18,24,27,38,28,39,26,29)(20,30,34,44,36,45,35,46)(25,37,40,47,41,48,42,43);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,42)(27,40)(28,41)(29,43)(31,46)(32,44)(33,45)(38,47)(39,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,39,38)(25,40,41,42)(30,44,45,46)(37,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100196);