#416416

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,22,23,32)(13,17)(14,18)(16,24,31,25)(19,34,35,39)(20,28)(21,29)(26,33)(27,36,30,38)(37,40);(9,11,13,21,14,20,26,37)(10,15,17,29,18,28,33,40)(12,19,25,38,23,35,24,36)(16,30,32,34,31,27,22,39);(2,5)(6,7)(9,12)(10,16)(11,19)(13,25)(14,23)(15,27)(17,32)(18,31)(20,35)(21,38)(22,33)(24,26)(28,30)(29,39)(34,40)(36,37);(1,3,4,8)(2,6,5,7)(11,20)(15,28)(19,35)(21,37)(27,30)(29,40)(34,39)(36,38)

Polycyclic presentation:

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

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, 323612);