#423664

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1597 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)(5,6)(7,8)(9,10)(11,20)(12,14)(13,26)(15,19)(16,22)(17,28)(18,24)(21,36)(23,35)(25,29)(27,33)(30,32)(31,38)(34,37)(39,40);(9,11,15,23)(10,16,19,32)(12,20,18,35)(13,27,29,39)(14,30,24,22)(17,33,34,40)(21,37,38,28)(25,36,26,31);(1,2,3,4)(9,12,24,19,15,18,14,10)(11,20,30,16,23,35,22,32)(13,25,37,34,29,26,28,17)(21,33,39,31,38,40,27,36);(6,8)(9,13,24,37,15,29,14,28)(10,17,12,25,19,34,18,26)(11,21,30,39,23,38,22,27)(16,31,35,40,32,36,20,33)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,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, 330860);