#6452028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2659 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)(3,8)(4,6)(5,7)(9,10,14,21,16,20,15,22)(11,23,26,34,28,35,27,17)(12,18,29,37,30,36,13,19)(24,33,25,38,31,40,39,32);(9,11)(10,17)(12,25)(13,31)(14,27)(15,26)(16,28)(18,33)(19,38)(20,34)(21,35)(22,23)(24,29)(30,39)(32,37)(36,40);(1,3,5,4)(2,6,7,8)(9,12,16,30)(10,18,20,36)(11,24,28,31)(13,14,29,15)(17,32,34,38)(19,21,37,22)(23,33,35,40)(25,27,39,26);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,25)(12,14)(15,30)(16,29)(17,33)(18,21)(20,37)(22,36)(23,38)(24,27)(26,31)(28,39)(32,35)(34,40);(10,20)(17,34)(18,36)(19,37)(21,22)(23,35)(32,38)(33,40)

Polycyclic presentation:

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

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, 6359224);