#6546723

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

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^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, 6453919);