#6344247

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	C₄²	C₂⁵
prime radical	#62835	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2786 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bf,d^a=dh,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, 6251443);