#6344202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
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	C₄²	C₂⁵
prime radical	#62838	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251398);