#418652

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 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)(9,10,13,19)(11,21,24,16)(12,26,28,33)(14,17,15,20)(18,29,35,30)(22,36,25,32)(23,38,27,40)(31,37,34,39);(1,2,4,5)(3,7,8,6)(9,11,13,24)(10,16,19,21)(12,27,28,23)(14,22,15,25)(17,32,20,36)(18,34,35,31)(26,38,33,40)(29,37,30,39);(10,17)(11,22)(14,15)(18,33)(19,20)(23,37)(24,25)(26,35)(27,39)(29,30)(32,36)(38,40);(1,3,4,8)(2,6,5,7)(9,12,13,28)(10,18,19,35)(11,23,24,27)(14,30,15,29)(16,31,21,34)(17,33,20,26)(22,37,25,39)(32,38,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,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, 325848);