#6499480

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#35303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6406676);