#6343654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
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	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2942 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=hi,f^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250850);