#6506401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(3,8)(4,6)(5,7)(17,18,24,28,23,29,22,39)(19,30,34,46,33,25,32,42)(20,27,35,43,21,36,38,26)(31,41,47,40,37,44,48,45);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,37)(22,32)(23,33)(24,34)(26,40)(27,44)(28,42)(29,30)(35,47)(36,41)(38,48)(39,46)(43,45);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,43)(19,31,33,37)(22,35,24,38)(25,40,30,45)(27,28,36,39)(32,47,34,48)(41,46,44,42);(11,16)(13,15)(17,21,23,20)(18,27,29,36)(19,31,33,37)(22,35,24,38)(25,41,30,44)(26,28,43,39)(32,48,34,47)(40,46,45,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,34)(20,35,21,38)(25,42,30,46)(26,36,43,27)(31,47,37,48)(40,41,45,44)

Polycyclic presentation:

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

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, 6413597);