#449759

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356955);