#6449729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6356925);