#459765

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,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, 366961);