#449803

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

1382 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)(6,7)(9,10)(11,20)(12,23,26,17)(13,16)(14,18)(15,30)(19,29)(21,38,25,32)(22,33)(24,34,39,40)(27,35,31,37)(28,36);(1,2,4,5)(3,7,8,6)(9,11)(10,16)(12,24)(13,14)(15,28)(17,34)(18,20)(19,36)(21,27)(22,29)(23,40)(25,31)(26,39)(30,33)(32,35)(37,38);(9,12)(10,17)(11,21)(13,25)(14,26)(15,31)(16,32)(18,23)(19,37)(20,38)(22,39)(24,28)(27,29)(30,35)(33,40)(34,36);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,18,20)(12,25,26,21)(15,28,29,22)(17,32,23,38)(19,33,30,36)(24,27,39,31)(34,37,40,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=e,g^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,g^b=gi,g^c=gi,h^c=hi,f^d=fi

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, 356999);