#411134

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318330);