#408634

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3062	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=f,e^2=i,f^2=i,g^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,g^b=gi,e^c=ei,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, 315830);