#6741566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,24,32,39)(20,25,33,40)(21,26,22,27)(29,36,31,38)(30,37,45,48)(34,41,35,42)(43,46,44,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,24,28,39)(20,30,33,45)(21,31,22,29)(25,37,40,48)(26,38,27,36)(34,44,35,43)(41,47,42,46);(2,6)(5,7)(10,14)(15,16)(19,29)(20,33)(21,22)(24,36)(25,40)(26,27)(30,44)(31,32)(37,47)(38,39)(43,45)(46,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,23,33)(18,25,28,40)(19,30,32,45)(21,34,22,35)(24,37,39,48)(26,41,27,42)(29,43,31,44)(36,46,38,47);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,21)(18,26)(20,34)(22,23)(25,41)(27,28)(29,31)(33,35)(36,38)(40,42)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648762);