#414684

order	512
nilpotency class	3
isoclinism	isoclinic class of #394038
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1552 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:

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

Polycyclic presentation:

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

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, 321880);