#6513280

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62888	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420476);