#420478

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1714 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 327674);