#417075

order	512
nilpotency class	3
isoclinism	isoclinic class of #396613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

728 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 324271);