#6460715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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)(19,27)(20,25)(21,23)(22,34)(24,31)(26,29)(28,40)(30,39)(32,36)(33,45)(35,42)(37,44)(38,43)(41,47)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,34)(31,42)(33,43)(37,46)(39,47);(10,14)(11,15)(17,19,29,42,21,28,22,31)(18,24,34,40,23,35,26,27)(20,30,43,48,32,41,33,44)(25,37,45,47,36,46,38,39);(11,15)(13,16)(17,20,21,32)(18,25,23,36)(19,30,28,41)(22,33,29,43)(24,37,35,46)(26,38,34,45)(27,39,40,47)(31,44,42,48);(1,3,4,7)(2,6,5,8)(19,28)(20,32)(24,35)(25,36)(27,40)(31,42)(33,43)(38,45)

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367911);