#6448648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
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	C₂⁴	C₂⁵
prime radical	#62844	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3446 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355844);