#6343173

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2783 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:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(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);(17,19,23,32)(18,24,28,39)(20,30,34,46)(21,31,22,36)(25,37,42,48)(26,29,27,38)(33,44,35,45)(40,47,41,43);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,41)(27,40)(28,42)(29,43)(31,45)(32,46)(36,44)(38,47)(39,48);(3,9)(6,12)(10,15)(13,16)(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);(2,7)(6,12)(8,14)(13,16)(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,b^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fh,d^c=dhi,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, 6250369);