#6516928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424124);