#6437697

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4634 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,33)(23,32)(25,40)(26,37)(27,36)(28,35)(29,38)(30,39)(31,34);(2,4)(9,11)(10,16)(12,26)(13,28)(14,24)(15,31)(17,35)(18,37)(19,21)(20,40)(22,27)(23,29)(25,30)(32,36)(33,38)(34,39);(6,8)(9,12,14,13)(10,17,19,18)(11,22,24,23)(15,29,30,27)(16,32,21,33)(20,38,39,36)(25,28,31,26)(34,37,40,35);(9,13,14,12)(10,18,19,17)(11,23,24,22)(15,27,30,29)(16,33,21,32)(20,36,39,38)(25,26,31,28)(34,35,40,37);(10,19)(11,24)(17,18)(20,39)(22,23)(25,31)(26,28)(36,38)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344893);