#470997

order	512
nilpotency class	3
isoclinism	isoclinic class of #460131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378193);