#193145

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100341);