#422886

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1883 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330082);