#6546650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1544 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)(22,37)(23,27)(24,31)(29,45)(30,40)(32,48)(33,42)(34,43)(35,39)(36,44)(38,47)(41,46);(9,10)(11,14)(12,13)(15,16)(17,19,21,35,23,30,22,31)(18,24,37,40,27,39,26,28)(20,32,36,47,33,29,34,46)(25,41,43,45,42,38,44,48);(2,5)(6,8)(10,13)(14,16)(17,20,23,33)(18,25,27,42)(19,29,30,32)(21,36,22,34)(24,38,39,41)(26,44,37,43)(28,45,40,48)(31,47,35,46);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(24,39)(25,42)(28,40)(31,35)(34,36)(43,44);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,35)(20,34,33,36)(24,40,39,28)(25,43,42,44)(29,46,32,47)(38,45,41,48)

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,e^b=ei,d^c=di,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453846);