#6574337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 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)(3,4)(9,10,15,19)(11,20,22,36)(12,17,26,33)(13,18,14,29)(16,23,31,24)(21,35,25,39)(27,34,28,40)(30,37,32,38);(5,6)(7,8)(9,11,13,23,15,22,14,24)(10,16,29,36,19,31,18,20)(12,25,27,38,26,21,28,37)(17,32,40,35,33,30,34,39);(2,4)(6,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,26)(16,30)(18,34)(19,33)(20,35)(22,25)(23,37)(24,38)(29,40)(31,32)(36,39);(5,6)(7,8)(11,22)(12,26)(16,31)(17,33)(20,36)(23,24)(27,28)(34,40);(6,8)(9,13,15,14)(10,18,19,29)(11,23,22,24)(12,27,26,28)(16,20,31,36)(17,34,33,40)(21,37,25,38)(30,35,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481533);