#6437666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4757 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)(5,6)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,28)(20,29)(21,31)(22,30)(23,32)(24,34)(25,33)(26,36)(27,35)(37,39)(38,40);(2,4)(9,11)(10,15)(12,23)(13,20)(14,22)(16,32)(17,30)(18,29)(19,24)(21,27)(25,38)(26,37)(28,34)(31,35)(33,40)(36,39);(6,8)(9,12,14,25)(10,16,17,33)(11,19,22,37)(13,26,27,24)(15,28,30,39)(18,36,35,34)(20,23,21,38)(29,32,31,40);(11,20)(13,27)(15,29)(18,35)(19,23)(21,22)(24,26)(28,32)(30,31)(34,36)(37,38)(39,40);(10,17)(15,30)(16,33)(18,35)(28,39)(29,31)(32,40)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315554, #72315560, #72315565, …
maximal normal subgroup of	#72315554, #72315560, #72315565, …
maximal quotient of	#72315658, #72315938, #72315952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344862);