#413576

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 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)(3,8,4,6)(9,10)(11,23)(12,28)(13,32)(14,21)(15,20)(16,22)(17,25)(18,30)(19,33)(24,40)(26,37)(27,36)(29,39)(31,35)(34,38);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,13)(14,26)(15,25)(16,27)(18,38)(19,35)(20,36)(21,23)(22,37)(24,30)(28,40)(29,33)(31,34)(32,39);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,24)(13,25)(14,30)(15,29)(16,31)(17,19)(20,28)(21,35)(22,39)(23,32)(26,34)(27,33)(36,38)(37,40);(9,13,15,33,16,34,14,24)(10,19,21,32,22,40,20,38)(11,12,25,29,27,31,26,30)(17,35,23,39,37,28,36,18)

Polycyclic presentation:

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

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, 320772);