#6546570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453766);