#6607943

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 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)(9,10,13,15)(11,16,12,14)(17,18,23,27)(19,24,29,39)(20,32,36,48)(21,26,22,37)(25,34,42,35)(28,43,33,47)(30,40,31,46)(38,44,41,45);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,29)(18,24,27,39)(20,33,36,28)(21,30,22,31)(25,41,42,38)(26,40,37,46)(32,47,48,43)(34,45,35,44);(11,12)(14,16)(17,20,21,34,23,36,22,35)(18,25,37,48,27,42,26,32)(19,28,30,44,29,33,31,45)(24,38,46,47,39,41,40,43);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,29)(24,39)(28,33)(30,31)(38,41)(40,46)(43,47)(44,45);(11,12)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,31)(20,34,36,35)(24,40,39,46)(25,32,42,48)(28,44,33,45)(38,43,41,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,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, 6515139);