#6729628

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6636824);