#6606008

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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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)(17,18,22,26)(19,27,30,44)(20,32,34,47)(21,36,28,25)(23,37,39,31)(24,40,42,35)(29,41,33,38)(43,48,46,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,28,37,22,30,21,31)(18,23,36,44,26,39,25,27)(20,33,40,45,34,29,35,48)(24,41,47,43,42,38,32,46);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,28,40,22,34,21,35)(18,24,36,47,26,42,25,32)(19,29,37,48,30,33,31,45)(23,38,44,46,39,41,27,43);(9,11)(10,13)(12,15)(14,16)(20,34)(24,42)(29,33)(32,47)(35,40)(38,41)(43,46)(45,48);(11,15)(13,16)(19,30)(23,39)(27,44)(29,33)(31,37)(38,41)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fi,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, 6513204);