#410837

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1357 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)(11,21)(12,26)(13,19)(14,18)(15,20)(16,24)(17,27)(22,30)(23,32)(25,31)(28,35)(29,34)(33,37)(36,38)(39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16,20,31)(12,22)(13,24)(14,25)(15,23)(17,33,34,30)(18,21,19,32)(26,36,35,40)(27,38)(28,37)(29,39);(9,12)(10,17,20,34)(11,22)(13,29)(14,27)(15,28)(16,30,31,33)(18,35,19,26)(21,36,32,40)(23,37)(24,39)(25,38);(2,6)(5,8)(11,23)(16,31)(21,32)(22,37)(24,25)(30,33)(36,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318033);