#6343683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343678
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	#62822	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2860 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)(3,8)(4,9)(5,10)(6,11)(7,12)(13,15)(14,16)(17,18)(19,29,33,24)(20,25)(21,26)(22,27)(23,28)(30,43,45,37)(31,39,32,38)(34,40)(35,41)(36,42)(44,48,46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,10)(9,15)(11,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,37)(26,38)(27,39)(28,29)(34,44)(35,45)(36,46)(40,47)(41,43)(42,48);(1,3)(2,8)(4,14)(5,12)(6,13)(7,10)(9,16)(11,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,37)(26,40)(27,42)(28,41)(29,43)(31,44)(32,46)(33,45)(38,47)(39,48);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,15,16,8)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,35,36)(24,38,29,39)(25,40,41,42)(30,44,45,46)(37,47,43,48);(1,4,6,5)(2,10,11,9)(3,12,13,14)(7,8,16,15)(17,22,23,21)(18,27,28,26)(19,32,33,31)(20,34,35,36)(24,39,29,38)(25,40,41,42)(30,44,45,46)(37,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250879);