#412122

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=di,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, 319318);