#6697266

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(10,14)(11,12)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,33)(28,38)(29,39)(32,41)(34,44)(35,43)(36,40)(37,42)(45,46)(47,48);(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,41,31,48)(26,39,43,40)(28,37,46,34)(38,44,45,42);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,35)(39,41)(40,48)(42,43);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(18,25)(20,33)(23,27)(26,43)(28,46)(32,47)(34,37)(39,40);(10,14)(15,16)(18,25)(19,29)(20,33)(22,35)(23,39)(27,40)(28,32)(30,36)(38,48)(41,45)(42,44)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6604462);