#6546580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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)(17,18,23,27)(19,28,30,45)(20,33,35,25)(21,26,22,38)(24,31,40,32)(29,46,34,44)(36,43,37,42)(39,47,41,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,40,38,45)(20,34,36,48,35,29,37,47)(25,41,43,46,33,39,42,44);(2,5)(6,8)(10,13)(14,16)(17,20,23,35)(18,25,27,33)(19,29,30,34)(21,37,22,36)(24,39,40,41)(26,43,38,42)(28,44,45,46)(31,48,32,47);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,35)(24,40)(25,33)(28,45)(31,32)(36,37)(42,43);(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,31,30,32)(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=f,c^2=i,d^2=gi,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=ehi

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, 6453776);