#411005

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

718 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:

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

Polycyclic presentation:

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

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, 318201);