#6506433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413629);