#6466879

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6374075);