#6422220

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2407	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,29,30,34,28,33)(27,35,39,40,38,32,36,37);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,40)(34,35);(2,6)(3,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,31)(26,33,34,29)(27,36,38,39)(32,40,35,37);(2,6)(5,8)(11,19)(14,17)(18,24)(21,23)(25,29,31,34,30,33,28,26)(27,37,39,35,38,40,36,32);(10,16)(14,17)(15,22)(21,23)(27,38)(32,35)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=dfi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6329416);