#6546747

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=gi,e^2=g,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, 6453943);