#6343720

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250916);