#6437677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4645 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)(11,16)(12,17)(13,18)(14,20)(15,19)(21,30)(22,32)(23,31)(24,33)(25,34)(26,35)(27,36)(28,38)(29,37)(39,40);(2,4)(9,11)(10,16)(12,24)(13,26)(14,28)(15,23)(17,33)(18,35)(19,31)(20,38)(21,25)(22,29)(27,39)(30,34)(32,37)(36,40);(6,8)(9,12,15,13)(10,17,19,18)(11,21,23,39)(14,27,29,25)(16,30,31,40)(20,36,37,34)(22,26,28,24)(32,35,38,33);(9,13,15,12)(10,18,19,17)(11,21,23,39)(14,25,29,27)(16,30,31,40)(20,34,37,36)(22,26,28,24)(32,35,38,33);(10,19)(16,31)(17,18)(20,37)(30,40)(32,38)(33,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315554, #72315560, #72315569
maximal normal subgroup of	#72315554, #72315560, #72315569
maximal quotient of	#72315737, #72315780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344873);