#6495490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(11,15)(14,16)(17,18,23,36,22,28,21,27)(19,29,35,24,32,43,31,40)(20,26,37,41,33,25,34,42)(30,45,48,39,46,44,47,38);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,24,28,40)(20,30,33,46)(21,35,23,31)(25,38,26,39)(27,43,36,29)(34,48,37,47)(41,45,42,44);(2,5)(6,8)(10,13)(14,16)(17,20,22,33)(18,25,28,26)(19,30,32,46)(21,34,23,37)(24,38,40,39)(27,41,36,42)(29,44,43,45)(31,47,35,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,26)(19,30)(21,34)(22,33)(23,37)(24,39)(25,28)(27,42)(29,45)(31,47)(32,46)(35,48)(36,41)(38,40)(43,44);(9,11,12,15)(10,14,13,16)(20,33)(25,26)(30,46)(34,37)(38,39)(41,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6402686);