#6406689

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6313885);