#6606003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,34)(21,25)(22,27)(23,31)(24,37)(26,29)(30,42)(32,41)(33,40)(35,39)(36,44)(38,43)(45,47)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,32,22,33,21,31)(18,23,25,40,27,41,26,28)(20,35,36,47,38,30,37,46)(24,42,43,45,44,39,34,48);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,29,36,22,38,21,37)(18,24,25,43,27,44,26,34)(19,30,32,46,33,35,31,47)(23,39,40,48,41,42,28,45);(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,25,27,26)(19,31,33,32)(20,36,38,37)(23,40,41,28)(24,34,44,43)(30,46,35,47)(39,45,42,48);(11,15)(13,16)(17,21,22,29)(18,25,27,26)(19,32,33,31)(20,37,38,36)(23,28,41,40)(24,43,44,34)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,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, 6513199);