#6629375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628867
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6536571);