#193366

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,e^c=eh,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, 100562);