#408082

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

740 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)(3,5)(4,6)(7,8)(9,10,15,21)(11,22)(12,27,30,18)(13,19,29,35)(14,20,16,31)(17,25)(23,34)(24,33)(26,36)(28,32)(37,40)(38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,23)(14,25)(15,26)(16,24)(18,34)(19,32)(20,36)(21,33)(22,31)(27,39)(29,38)(30,37)(35,40);(9,12,14,29,15,30,16,13)(10,18,20,19,21,27,31,35)(11,23,24,37,26,38,25,28)(17,32,22,34,33,40,36,39);(2,6)(5,8)(9,13,16,30,15,29,14,12)(10,19,31,18,21,35,20,27)(11,23,24,37,26,38,25,28)(17,32,22,34,33,40,36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 315278);