#420821

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1896 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:

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

Polycyclic presentation:

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

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, 328017);