#189315

order	512
nilpotency class	3
isoclinism	isoclinic class of #189312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,28)(20,33,35,40)(21,36)(22,26)(23,27)(24,31)(25,34,41,37)(29,38,44,48)(30,43)(32,39)(42,45,47,46);(2,7)(8,15)(9,10)(11,16)(17,19,21,31,23,30,22,32)(18,24,26,28,27,39,36,43)(20,29,34,45,35,44,37,46)(25,38,40,42,41,48,33,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,40)(27,41)(28,42)(30,44)(31,45)(32,46)(33,36)(39,48)(43,47);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(19,30)(24,39)(28,43)(29,44)(31,32)(38,48)(42,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^c=fhi,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, 96511);