#6437693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4710 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)(11,19)(12,16)(13,17)(14,18)(15,22)(20,37)(21,31)(23,30)(24,40)(25,33)(26,34)(27,35)(28,36)(29,38)(32,39);(2,4)(9,11)(10,15)(12,24)(13,22)(14,21)(16,32)(17,19)(18,30)(20,26)(23,27)(25,39)(28,38)(29,34)(31,35)(33,40)(36,37);(6,8)(9,12,13,25)(10,16,17,33)(11,20,22,38)(14,28,27,26)(15,29,19,37)(18,36,35,34)(21,24,23,39)(30,32,31,40);(11,21)(14,27)(15,30)(18,35)(19,31)(20,24)(22,23)(26,28)(29,32)(34,36)(37,40)(38,39);(10,17)(11,22)(16,33)(18,35)(20,38)(21,23)(24,39)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315582, #72315588
maximal normal subgroup of	#72315582, #72315588
maximal quotient of	#72315984

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344889);