#6729571

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(18,23)(19,27)(20,31)(26,34)(30,35)(32,45)(38,47)(43,48);(1,3)(2,5)(4,8)(6,7)(17,18,21,23)(19,28,27,41)(20,25,31,37)(22,34,33,26)(24,35,36,30)(29,43,42,48)(32,39,45,40)(38,46,47,44);(10,14)(15,16)(17,19,21,27)(18,24,23,36)(20,29,31,42)(22,35,33,30)(25,38,37,47)(26,41,34,28)(32,46,45,44)(39,43,40,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,31)(22,32)(23,37)(24,38)(26,40)(27,42)(28,43)(30,44)(33,45)(34,39)(35,46)(36,47)(41,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,26)(19,27)(20,31)(22,33)(23,34)(24,28)(25,39)(36,41)(37,40)(38,48)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6636767);