#6437426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6174 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,35)(23,30)(24,32)(25,33)(26,34)(28,31)(29,36)(37,40)(38,39);(5,6)(7,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,36)(25,37)(26,38)(32,39)(33,35)(34,40);(2,4)(6,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,26)(16,29)(18,32)(19,34)(20,35)(22,36)(23,38)(27,33)(28,37)(30,39)(31,40);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,30,31,20)(17,32,34,33)(21,36,38,37)(29,39,40,35);(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,24)(16,30,31,20)(17,33,34,32)(21,37,38,36)(29,35,40,39)

Polycyclic presentation:

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

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, 6344622);