#6407830

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1307 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,31,36,30,35,29,34)(27,33,37,40,28,32,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,32)(28,30)(29,38)(31,37)(33,35)(34,40)(36,39);(2,5)(6,7)(11,18)(14,17)(19,24)(21,23)(25,27)(26,32)(28,30)(29,38)(31,37)(33,35)(34,40)(36,39);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,22)(19,24)(25,28,30,27)(26,33,35,32)(29,37,31,38)(34,39,36,40);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(25,29,30,31)(26,34,35,36)(27,37,28,38)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfg,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=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, 6315026);