#193013

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

811 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)(4,7)(5,9)(6,10)(8,13)(12,16)(14,15)(17,18,22,26,23,27,21,35)(19,28,30,24,31,38,36,39)(20,25,32,41,34,42,33,40)(29,43,45,37,46,47,44,48);(2,7)(4,5)(8,15)(9,10)(11,16)(12,13)(17,19,22,30,23,31,21,36)(18,24,26,38,27,39,35,28)(20,29,32,45,34,46,33,44)(25,37,41,47,42,48,40,43);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,15,9,11)(17,20,23,34)(18,25,27,42)(19,29,31,46)(21,33,22,32)(24,37,39,48)(26,41,35,40)(28,43,38,47)(30,45,36,44);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,11,16,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,28,39,38)(25,40,42,41)(29,44,46,45)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,d^b=dh,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, 100209);