#420121

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327317);